Differentiating power series

WEBVTT
Kind: captions
Language: en

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- [Instructor] We're told here that

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f(x) is equal to this infinite series,

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and we need to figure out what is

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the third derivative of f,
evaluated at x equals zero.

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And like always, pause this
video and see if you can

00:00:12.026 --> 00:00:15.493
work it out on your own
before we do it together.

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Alright, so there's two
ways to approach this.

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One is we could just take the derivative

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of this expression while
it's in sigma notation.

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The other way you do it is we could just

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expand out f(x) and take
the derivative three times,

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and see if we get an answer
that, I guess, makes sense.

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Let me do it the second way first.

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Let me just expand it out.

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F(x) is equal to, let's see,
when n is equal to zero,

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this is negative one to the
zero, which is just one,

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times x, times x to the zero plus three,

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so it's gonna be x to the third

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over two times zero, so that's
zero plus one factorial,

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so that's just going to be over one.

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Then the next term,
when n is equal to one,

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well now it's gonna be
negative one to the one,

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so now we're just gonna
have a negative out front.

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Negative, and it's gonna be
two times one plus three,

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so that's gonna be x to the fifth power

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over two times one plus
one, so it's gonna be

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two plus one is three factorial.

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So it's gonna be x to the fifth over six.

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And then when x is equal to two,

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this is going to be positive again,

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and it's gonna be x to the seventh power

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over five factorial.

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Is that right? Yeah.

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Five factorial, and five factorial...

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Actually, let me just write
that out as five factorial.

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Five factorial would be,
what, it would be 120.

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It'd be five times four
times six, so it'd be 120.

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But we could just keep going minus plus,

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and it goes on and on and on forever.

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Well now let's just take the derivatives.

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F'(x) is going to be equal to,

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we're still applying the power rule here,

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it's going to be three x squared

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minus 5/6x to the fourth, plus seven

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over five factorial x to the sixth,

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I'm just applying the power rule,

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minus plus, we just keep going
on and on and on forever.

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The second derivative, f''(x) is going

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to be equal to, apply
the power rule again.

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It's going to be six x to the
first minus four times five

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over six, I'll just write
that as 20/6x to the third,

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plus six times seven, so
it's 42 over five factorial

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x to the fifth, and we're
just gonna keep on going,

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minus plus, keep going,
or alternate between minus

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something then plus something,
on and on and on forever.

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Then we get to the third derivative.

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The third derivative is equal to,

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let's see, the derivative of six x is six,

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and then we have minus
20 times three is 60/6,

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which of course is 10, x squared,

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plus five times 42 is what,
210 over five factorial

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times x to the fourth power, minus plus

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over and over and over again, and then

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we just evaluate this at zero.

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F'''(0), well, when x is equal to zero,

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all of these terms with
xes are gonna go to zero,

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and you're just gonna be
left with this six here.

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So f'', the third derivative evaluated

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at zero is just equal to six.

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Now another way that we
could've tackled this

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is just kept it in this sigma notation.

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We could've said that f'(x) is equal

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to the infinite sum, and
actually, let me line them up.

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So this is where we
did f'(x) expanded out,

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but we could've said
f'(x) is equal to the sum

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from n equals zero to infinity,

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and you take the derivative here,

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you're gonna get, and
you're taking the derivative

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with respect to x, so for that purpose,

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you assume everything else is,

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well, the n is just gonna tell us,

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is gonna tell us how we
change from term to term,

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so if we take the derivative
with respect to x here,

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use the power rule, bring the
two n plus three out front,

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so it's gonna be negative one to the n

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times two n plus three,
times x to the decrement,

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the exponent, two n plus two over

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two n plus one factorial.

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And then if you wanna take
the second derivative,

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and this is the same thing as this.

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If you take the second derivative, f''(x),

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well now we're taking the sum from zero

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to infinity of negative one to the n.

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Let me move over to the right a little bit

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so we have some space.

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And now, we take this exponent out front,

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so you're gonna have two n plus three

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times two n plus two,
all of that's going to be

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over two n plus one factorial,
and this is gonna be

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times x to the two n plus one.

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All I'm doing every time,
it seems really complicated,

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I'm just taking the exponent out front,

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multiplying it out front,
and then decrementing it.

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So two n plus two minus
one is two n plus one.

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And then if I wanna take
the third derivative,

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the third derivative is
the sum where n equals zero

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to infinity, negative one to the n.

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We take this, bring it, multiply it,

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so we're gonna have two n plus three

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times two n plus two,
times two n plus one,

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all of that over two n plus one factorial,

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and then that is going to be times

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x to the two n power.

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Now let's now evaluate this
thing when x is equal to zero.

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F'''(0) is gonna be the
sum from n equals zero

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to infinity of negative
one to the nth power.

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This is gonna be interesting.

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We're gonna have all of this business,

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two n plus three times two n plus two

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times two n plus one, all of that

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over two n plus one factorial,

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times zero to the two n power.

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You might be tempted to say, well, hey,

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if zero to these different powers,

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maybe everything's gonna be zero,

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but remember, we're
starting at n equals zero,

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so for any n that's not equal to zero,

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this zero to that power
is just gonna be zero

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and that term's gonna disappear,

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kinda like what we saw
when we expanded it out.

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And so the only term that matters

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here is when n is equal to zero.

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So this is just going to be equal to,

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because for n equals one,
two, three, four, five,

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all the way to infinity,
this thing is gonna dominate.

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It's just gonna multiply,
it's gonna be zero.

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You're just gonna zero everything out.

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And so this is just gonna
reduce to the first term,

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when n equals zero,
and when n equals zero,

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it's gonna be negative one to the zero.

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This is gonna be, which is just one.

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Let me just write that as one.

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Times, this is three times two times one

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over one factorial, and
then times zero to the zero,

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which is equal to one.

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So this is equal to one,
and so this is equal to six.

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Either way, I think
the first way we did it

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was a little bit more straightforward,

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a little bit more
intuitive, closer to what

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you might have seen
before, but it's important

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to realize that we did
the same thing both times,

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we just kept it in the sigma notation

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on this time to the right.

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This technique is useful
because you'll see it a lot

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in math, where you might
wanna do things a little

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bit more of a general way,
and so it might be helpful

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to take the derivatives while you stay

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in that sigma notation.


Integrating power series

WEBVTT
Kind: captions
Language: en

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- [Instructor] So we're told that f(x)

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is equal to the infinite series,

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we're going from n equals one to infinity

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of n plus one over four to the
n plus one, times x to the n.

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And what we wanna figure out is, what is

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the definite integral from
zero to one of this f(x)?

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And like always, if you feel inspired,

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and I encourage you to feel inspired,

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pause the video and see if
you can work through this

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on your own, or at any time
while I'm working through it,

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pause it and try to keep on going.

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Well, let's just rewrite
this a little bit.

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This is going to be the same thing

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as the integral from zero to one.

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F(x) is this series, so I could write

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the sum from n equals one to infinity

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of n plus one over four to the n plus one,

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times x to the n.

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And now what I'm about
to do might be the thing

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that might be new to some of you,

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but this is essentially, we're taking

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a definite integral of a sum of terms.

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That's the same thing as taking the sum

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of a bunch of definite integrals.

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Let me make that clear.

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So if I had a, let's say
this is a definite integral

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zero to one, and let's say
I had a bunch of terms here.

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I could even call them functions.

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Let's say it was g(x) plus h(x),

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and I just kept going
on and on and on, dx,

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well, this is the same thing
as a sum of the integrals,

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as the integral from zero to one of g(x),

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g(x) dx plus the integral
from zero to one h(x) dx,

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plus, and we go on and on and on forever.

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However many of these terms are.

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This comes straight out of
our integration properties.

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We can do the exact same thing here,

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although we'll just do it
with the sigma notation.

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This is going to be equal to the sum

00:01:49.095 --> 00:01:51.512
from n equals one to infinity

00:01:52.562 --> 00:01:56.446
of the definite integral
of each of these terms.

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So I'm gonna write it like this.

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So of the integral from zero to one

00:02:01.962 --> 00:02:06.129
of n plus one over four
to the n plus oneth power,

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times x to the n, and then it is dx.

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Once again, now we're taking
the sum of each of these terms.

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Let's evaluate this
business right over here.

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That is going to, I'll
just keep writing it out.

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This is going to be equal to the sum

00:02:28.564 --> 00:02:32.231
from n equals one to
infinity, and then the stuff

00:02:32.231 --> 00:02:33.832
that I just underlined in orange,

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this is going to be, let's see,

00:02:34.665 --> 00:02:37.530
we take the antiderivative here.

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We are going to get to
x to the n plus one,

00:02:42.513 --> 00:02:44.263
and then we're gonna divide by n plus one.

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So we have this original n plus one over

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four to the n plus one,
and that's just a constant

00:02:52.280 --> 00:02:54.464
when we think in terms of x, for any one

00:02:54.464 --> 00:02:57.053
of these terms, and then
here we'd wanna increment

00:02:57.053 --> 00:02:59.997
the exponent, and then divide
by that incremented exponent.

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This just comes out of, I
often call it the inverse,

00:03:02.446 --> 00:03:06.945
or the anti-power rule, or
reversing the power rule.

00:03:06.945 --> 00:03:10.862
So it's x to the n plus
one over n plus one.

00:03:12.294 --> 00:03:13.829
I just took the antiderivative,

00:03:13.829 --> 00:03:17.145
and we're gonna go from zero
to one for each of these terms.

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Before we do that, we can simplify.

00:03:18.444 --> 00:03:21.511
We have an n plus one,
we have an n plus one,

00:03:21.511 --> 00:03:23.927
and so we can rewrite all of this.

00:03:23.927 --> 00:03:25.743
This is going to be the same thing,

00:03:25.743 --> 00:03:29.177
we're gonna take the sum from
n equals one to infinity,

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and this is going to be,
what we have in here,

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when x is equal to one, it is one,

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we could write one to the n plus one

00:03:40.123 --> 00:03:42.494
over four to the n plus one.

00:03:42.494 --> 00:03:44.160
Actually, yeah, why don't
I write it that way.

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One to the n plus one over
four to the n plus one,

00:03:48.443 --> 00:03:51.693
minus zero to the n plus one
over four to the n plus one,

00:03:51.693 --> 00:03:53.576
so we're not gonna even
have to write that.

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I could write zero to the n plus one

00:03:55.626 --> 00:04:00.566
over four to the n plus one,
but this is clearly just zero.

00:04:00.566 --> 00:04:02.509
And then this, and this is starting

00:04:02.509 --> 00:04:05.161
to get nice and simple now, this is going

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to be the same thing, this is equal

00:04:06.958 --> 00:04:10.375
to the sum from n equals one to infinity.

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And we almost are gonna
get to our drumroll

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of 1/4 to the n plus one.

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Now you might immediately recognize this.

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This is an infinite geometric series.

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What is the first term here?

00:04:22.758 --> 00:04:25.258
Well, the first term is, well,

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when n is equal to one,
the first term here

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is 1/4 to the second power.

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Did I do that right?

00:04:40.674 --> 00:04:41.507
Yeah.

00:04:41.507 --> 00:04:45.241
When n is equal to one, it's going to be,

00:04:45.241 --> 00:04:49.505
so this is going to be
1/4 to the second power,

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which is equal to 1/16,
so that's our first term.

00:04:54.425 --> 00:04:58.592
And then our common ratio
here, well that's gonna be,

00:05:02.029 --> 00:05:04.539
well, we're just gonna
keep multiplying by 1/4,

00:05:04.539 --> 00:05:06.888
so our common ratio here is 1/4.

00:05:06.888 --> 00:05:09.121
And so for an infinite geometric series,

00:05:09.121 --> 00:05:11.788
this is, since our common ratio,

00:05:14.063 --> 00:05:16.107
its absolute value is less than one,

00:05:16.107 --> 00:05:17.890
we know that this is going to converge,

00:05:17.890 --> 00:05:19.719
and it's gonna converge to the value,

00:05:19.719 --> 00:05:23.302
our first term, 1/16, divided by

00:05:24.718 --> 00:05:28.589
one minus the common ratio, one minus 1/4,

00:05:28.589 --> 00:05:32.756
so this is 3/4, so it's
equal to 1/16 times 4/3.

00:05:37.545 --> 00:05:40.858
This is going to be equal to 1/12.

00:05:40.858 --> 00:05:41.946
And we're done.

00:05:41.946 --> 00:05:43.725
And this seemed really daunting at first,

00:05:43.725 --> 00:05:45.099
but we just have to realize, okay,

00:05:45.099 --> 00:05:48.258
an integral of a sum,
even an infinite sum,

00:05:48.258 --> 00:05:50.975
well, that's gonna be the sum
of these infinite integrals.

00:05:50.975 --> 00:05:53.474
We take the antiderivative
of these infinite integrals,

00:05:53.474 --> 00:05:54.524
which we were able to do, which is kind

00:05:54.524 --> 00:05:58.091
of a cool thing, one of the
powers of symbolic mathematics,

00:05:58.091 --> 00:06:00.008
and then we realized, oh, we just have

00:06:00.008 --> 00:06:00.841
an infinite geometric series,

00:06:00.841 --> 00:06:02.467
which we know how to find the sum of.

00:06:02.467 --> 00:06:03.717
And we're done.


Worked example: alternating series

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:01.591
- [Instructor] What
are all positive values

00:00:01.591 --> 00:00:05.009
of p such that the series converges?

00:00:05.009 --> 00:00:05.842
So let's see.

00:00:05.842 --> 00:00:06.721
We have the sum from n equals one

00:00:06.721 --> 00:00:08.837
to infinity of negative one to the n

00:00:08.837 --> 00:00:11.814
plus one, times p over six to the n.

00:00:11.814 --> 00:00:13.489
So there's a couple of things
that might jump out at you.

00:00:13.489 --> 00:00:16.305
This negative one to the n plus one,

00:00:16.305 --> 00:00:18.771
as n goes from one to two to three,

00:00:18.771 --> 00:00:21.066
this is just going to
alternate between positive one,

00:00:21.066 --> 00:00:23.142
negative one, positive one, negative one.

00:00:23.142 --> 00:00:25.499
So we're gonna have alternating signs.

00:00:25.499 --> 00:00:27.417
That might be a little bit
of a clue of what's going on,

00:00:27.417 --> 00:00:28.759
and actually, let's just write it out.

00:00:28.759 --> 00:00:30.834
This is going to be,
see when n equals one,

00:00:30.834 --> 00:00:32.618
this is going to be to the second power,

00:00:32.618 --> 00:00:34.819
so it's gonna be positive one.

00:00:34.819 --> 00:00:37.867
So it's gonna be p over
six, and when n is two,

00:00:37.867 --> 00:00:39.450
this is gonna be to the third power,

00:00:39.450 --> 00:00:42.950
so it's gonna be minus p over six squared,

00:00:45.266 --> 00:00:49.200
then plus p over six to the third power.

00:00:49.200 --> 00:00:51.816
And I could even write to the
first power right over here.

00:00:51.816 --> 00:00:55.566
Then minus p over six to the fourth power,

00:00:55.566 --> 00:00:57.365
and we're gonna just
keep going plus, minus,

00:00:57.365 --> 00:01:00.387
on and on and on and on forever.

00:01:00.387 --> 00:01:03.857
This is clearly, this is a
classic alternating series

00:01:03.857 --> 00:01:05.623
right over here, and we could actually

00:01:05.623 --> 00:01:08.187
apply our alternating series test.

00:01:08.187 --> 00:01:10.454
And our alternating series test tells us

00:01:10.454 --> 00:01:13.273
that if this part of our expression,

00:01:13.273 --> 00:01:16.438
the part that is not alternating in sign,

00:01:16.438 --> 00:01:18.189
I guess you could say, if this part

00:01:18.189 --> 00:01:22.022
of the expression is
monotonically decreasing,

00:01:24.840 --> 00:01:26.854
monotonically decreasing,

00:01:26.854 --> 00:01:28.753
which is just a fancy way of saying that

00:01:28.753 --> 00:01:32.195
each successive term is less
than the term before it,

00:01:32.195 --> 00:01:34.678
and if we also know that the limit of this

00:01:34.678 --> 00:01:38.994
as n approaches infinity, that
also has to be equal to zero.

00:01:38.994 --> 00:01:41.961
So the limit as n approaches infinity

00:01:41.961 --> 00:01:46.128
of p over six to the nth power
also has to be equal to zero.

00:01:48.560 --> 00:01:51.779
Under what conditions is
that going to be true?

00:01:51.779 --> 00:01:54.011
Well, to meet either
one of those conditions,

00:01:54.011 --> 00:01:57.294
p over six has to be less than one.

00:01:57.294 --> 00:01:59.977
If p over six was equal to one,

00:01:59.977 --> 00:02:02.266
if, for example, p was six, well then

00:02:02.266 --> 00:02:04.010
we wouldn't be monotonically decreasing.

00:02:04.010 --> 00:02:06.360
Every term here would just be one.

00:02:06.360 --> 00:02:09.442
It would be one to the one,
one squared, on and on and on.

00:02:09.442 --> 00:02:12.826
And if p is greater than
six, well, then every time

00:02:12.826 --> 00:02:14.643
we multiply by p over six again,

00:02:14.643 --> 00:02:16.710
we would get a larger
number over and over again,

00:02:16.710 --> 00:02:19.309
and the limit for sure
would not be equal to zero.

00:02:19.309 --> 00:02:23.513
We could say p over six
needs to be less than one,

00:02:23.513 --> 00:02:25.650
and so multiply both sides by six

00:02:25.650 --> 00:02:28.509
and you get p needs to be less than six.

00:02:28.509 --> 00:02:32.868
And they told us, for what are
all the positive values of p?

00:02:32.868 --> 00:02:35.685
So we also know that p has
to be greater than zero.

00:02:35.685 --> 00:02:38.535
P is greater than zero and less than six,

00:02:38.535 --> 00:02:41.435
which is that choice right over here.

00:02:41.435 --> 00:02:43.685
Once again, we're not gonna
say less than or equal to six

00:02:43.685 --> 00:02:45.218
because if p was equal to six, this term

00:02:45.218 --> 00:02:47.902
is gonna be one to the
n, and so we're just

00:02:47.902 --> 00:02:49.668
gonna have, this would be one.

00:02:49.668 --> 00:02:51.535
It would be one minus one plus one,

00:02:51.535 --> 00:02:53.086
on and on and on and on forever.

00:02:53.086 --> 00:02:55.864
So definitely like that first choice.


Proof of p-series convergence criteria

WEBVTT
Kind: captions
Language: en

00:00:00.005 --> 00:00:01.510
- [Instructor] You might
recognize what we have

00:00:01.510 --> 00:00:05.978
here in yellow as the
general form of a p-Series,

00:00:05.978 --> 00:00:07.622
and what we're going
to do in this video is

00:00:07.622 --> 00:00:09.551
think about under which conditions,

00:00:09.551 --> 00:00:13.455
for what 'P's will this p-Series converge.

00:00:13.455 --> 00:00:15.701
And for it to be a p-Series, by definition

00:00:15.701 --> 00:00:18.534
P is going to be grater than zero.

00:00:19.474 --> 00:00:21.348
So I've set up some
visualizations to think

00:00:21.348 --> 00:00:23.950
about how we are going to understand when

00:00:23.950 --> 00:00:26.057
this p-Series converges.

00:00:26.057 --> 00:00:28.708
So over here you have the graph,

00:00:28.708 --> 00:00:31.468
this curve right here,
that's Y is equal to

00:00:31.468 --> 00:00:34.654
one over X to the P.

00:00:34.654 --> 00:00:36.092
And we're saying it in general terms,

00:00:36.092 --> 00:00:38.078
because P is greater
than zero, we know it's

00:00:38.078 --> 00:00:40.904
going to be a decreasing
function like this.

00:00:40.904 --> 00:00:43.274
Once again that's Y is equal to one over

00:00:43.274 --> 00:00:45.104
X to the P.

00:00:45.104 --> 00:00:47.271
And now what we've
shaded in ahead of time,

00:00:47.271 --> 00:00:48.989
underneath that curve, above the positive

00:00:48.989 --> 00:00:51.975
X axis, that's is the integral.

00:00:51.975 --> 00:00:54.538
From one to infinity,

00:00:54.538 --> 00:00:57.313
the improper integral
of one over X to the P,

00:00:57.313 --> 00:01:00.123
dX, so that's this area, that I've

00:01:00.123 --> 00:01:01.979
already shaded in, you see it in white

00:01:01.979 --> 00:01:03.975
in both of these graphs.

00:01:03.975 --> 00:01:05.422
And what we're going to hopefully

00:01:05.422 --> 00:01:08.150
see visually, is that there's a very close

00:01:08.150 --> 00:01:10.424
convergence or divergence
relationship between

00:01:10.424 --> 00:01:13.540
this p-Series, and this
integral, right over here.

00:01:13.540 --> 00:01:17.170
Because when we look at
this left hand graph,

00:01:17.170 --> 00:01:19.809
we see that this p-Series can be viewed

00:01:19.809 --> 00:01:23.729
as an upper Riemann
approximation of that area.

00:01:23.729 --> 00:01:25.239
What do I mean by that?

00:01:25.239 --> 00:01:27.132
Well think about

00:01:27.132 --> 00:01:30.549
the area of this first rectangle.

00:01:31.614 --> 00:01:34.614
The width is one, and its' height is

00:01:35.480 --> 00:01:40.002
one over one over one to the P.

00:01:40.002 --> 00:01:41.885
So this would be the first term,

00:01:41.885 --> 00:01:43.217
in this p-Series,

00:01:43.217 --> 00:01:45.553
this would just be an area of one.

00:01:45.553 --> 00:01:48.532
It's just the X and Y
scales are not the same.

00:01:48.532 --> 00:01:49.717
This one right over here;

00:01:49.717 --> 00:01:52.365
its' area would be one over two to the P.

00:01:52.365 --> 00:01:54.736
The area's one over three to the P.

00:01:54.736 --> 00:01:58.001
So the sum of the areas
of these rectangles,

00:01:58.001 --> 00:02:00.376
that is what this p-Series is, and

00:02:00.376 --> 00:02:02.565
you can see that each of these rectangles

00:02:02.565 --> 00:02:06.403
they are covering more than
the area under the curve.

00:02:06.403 --> 00:02:08.122
And so we know the area under the curve,

00:02:08.122 --> 00:02:09.701
that's gonna be greater than zero,

00:02:09.701 --> 00:02:13.505
this p-Series is going
to be greater than this

00:02:13.505 --> 00:02:16.079
integral, greater than
the area under the curve.

00:02:16.079 --> 00:02:18.515
But if we add one to the
area under the curve,

00:02:18.515 --> 00:02:20.883
now we're not just talking
about the white area,

00:02:20.883 --> 00:02:23.671
we're also talking about
this red area here,

00:02:23.671 --> 00:02:25.626
then our p-Series is going
to be less than that.

00:02:25.626 --> 00:02:29.118
Because the first term of
our p-Series is equal to one,

00:02:29.118 --> 00:02:31.168
and then all of the other terms,

00:02:31.168 --> 00:02:33.866
you can view it as a lower
Riemann approximation

00:02:33.866 --> 00:02:34.699
of the curve.

00:02:34.699 --> 00:02:36.756
And you can see, they fit under the curve,

00:02:36.756 --> 00:02:38.327
and they leave some area.

00:02:38.327 --> 00:02:41.508
So this is gonna be less
than that expression there.

00:02:41.508 --> 00:02:42.961
Now think about what happens;

00:02:42.961 --> 00:02:43.794
if we know

00:02:45.090 --> 00:02:48.477
that this right over here diverges,

00:02:48.477 --> 00:02:50.822
so if this improper integral diverges, it

00:02:50.822 --> 00:02:53.520
doesn't converge to a finite value,

00:02:53.520 --> 00:02:55.614
well the p-Series is greater than that,

00:02:55.614 --> 00:02:58.550
so if this diverges, then
that's going to diverge.

00:02:58.550 --> 00:03:01.955
Similarly, if this converges, the same

00:03:01.955 --> 00:03:03.131
integral right over here,

00:03:03.131 --> 00:03:05.524
if this converges, it
goes to a finite value,

00:03:05.524 --> 00:03:08.520
well one plus that is
still going to converge,

00:03:08.520 --> 00:03:11.227
so this, or p-Series must also converge,

00:03:11.227 --> 00:03:13.644
it must go to a finite value.

00:03:15.028 --> 00:03:16.660
All I'm talking about right here, this

00:03:16.660 --> 00:03:19.054
is really just the integral
test, when we think

00:03:19.054 --> 00:03:21.213
about tests of convergence and divergence,

00:03:21.213 --> 00:03:23.805
but I'm just making
sure that we have a nice

00:03:23.805 --> 00:03:25.818
conceptual understanding, and not just

00:03:25.818 --> 00:03:28.509
blindly applying the integral test.

00:03:28.509 --> 00:03:30.419
And you could go the other way too,

00:03:30.419 --> 00:03:33.348
if the p-Series converges,
then for sure this

00:03:33.348 --> 00:03:36.033
integral is going to converge, and if the

00:03:36.033 --> 00:03:38.165
p-Series diverges, then for sure

00:03:38.165 --> 00:03:39.658
this expression right
over here is going to

00:03:39.658 --> 00:03:42.201
diverge and the integral diverges.

00:03:42.201 --> 00:03:45.145
So we can say the p-Series converges if,

00:03:45.145 --> 00:03:48.734
and only if, this integral right over here

00:03:48.734 --> 00:03:49.971
converges.

00:03:49.971 --> 00:03:52.439
So, figuring out under what conditions

00:03:52.439 --> 00:03:55.139
for what P does the p-Series converge,

00:03:55.139 --> 00:03:56.963
it's boiling down to under what conditions

00:03:56.963 --> 00:03:59.228
does this integral converge?

00:03:59.228 --> 00:04:02.354
So let's scroll on down to
give us some real estate

00:04:02.354 --> 00:04:04.537
to think about what
has to be true for that

00:04:04.537 --> 00:04:06.324
integral to converge.

00:04:06.324 --> 00:04:08.094
So I'm gonna re-write it.

00:04:08.094 --> 00:04:10.357
So we've got the integral
from one to infinity,

00:04:10.357 --> 00:04:14.107
improper integral, of
one over X to the P, dX

00:04:15.153 --> 00:04:18.321
this is the same thing, this is the limit

00:04:18.321 --> 00:04:21.162
as, I'll use the variable
M since we're already

00:04:21.162 --> 00:04:24.427
using N, as M approaches infinity,

00:04:24.427 --> 00:04:26.388
and the integral from

00:04:26.388 --> 00:04:30.278
one to M of one over,

00:04:30.278 --> 00:04:33.188
and actually just let me
write that as X to the -P.

00:04:33.188 --> 00:04:36.500
X to the negative P, dX.

00:04:36.500 --> 00:04:37.933
And let me just focus on this, and we'll

00:04:37.933 --> 00:04:39.167
just remember that we're
gonna have to take the

00:04:39.167 --> 00:04:41.379
limit as M approaches infinity

00:04:41.379 --> 00:04:42.721
I don't wanna have to keep writing that

00:04:42.721 --> 00:04:44.719
over and over again.

00:04:44.719 --> 00:04:46.016
So let's think about what this is,

00:04:46.016 --> 00:04:48.849
so there's a couple of conditions;

00:04:48.849 --> 00:04:51.669
we know that P is greater than zero,

00:04:51.669 --> 00:04:54.092
but there's two situations
right over here:

00:04:54.092 --> 00:04:57.171
there's one situation
when P is equal to one.

00:04:57.171 --> 00:04:59.353
If P is equal to one,
then this is just the

00:04:59.353 --> 00:05:02.273
integral of one over X, and so this thing

00:05:02.273 --> 00:05:05.523
is going to be the integral of ln of X,

00:05:06.879 --> 00:05:10.005
and we're gonna go from one to M,

00:05:10.005 --> 00:05:13.690
and so this would be the natural log, of M

00:05:13.690 --> 00:05:15.970
minus the natural log of one, well E

00:05:15.970 --> 00:05:18.083
to the zero power is
one, so the natural log,

00:05:18.083 --> 00:05:18.916
I'll write it out,

00:05:18.916 --> 00:05:20.599
the natural log of one, but the natural

00:05:20.599 --> 00:05:22.014
log of one is just zero,

00:05:22.014 --> 00:05:24.758
so in this special case,
I guess we can say,

00:05:24.758 --> 00:05:26.007
when P equals one,

00:05:26.007 --> 00:05:29.068
this integral, from one to M, comes down

00:05:29.068 --> 00:05:31.066
to the natural log of M.

00:05:31.066 --> 00:05:33.202
Now let's think about the situation

00:05:33.202 --> 00:05:37.247
where P does not equal one,

00:05:37.247 --> 00:05:38.809
well there we're just
kind of reversing the

00:05:38.809 --> 00:05:40.524
power rule that we learned in basic

00:05:40.524 --> 00:05:41.627
differentiation.

00:05:41.627 --> 00:05:43.442
So we'd increment that exponent,

00:05:43.442 --> 00:05:47.071
so it'd be X to the negative P plus one,

00:05:47.071 --> 00:05:49.115
and we can even write that as

00:05:49.115 --> 00:05:52.233
X to the one minus P,
that's the same thing

00:05:52.233 --> 00:05:54.143
as negative P plus one,

00:05:54.143 --> 00:05:55.757
then we would divide by that.

00:05:55.757 --> 00:05:57.090
So one, minus P.

00:06:00.659 --> 00:06:04.326
We are gonna go from one to M,

00:06:04.326 --> 00:06:06.229
and so this is going to be equal to,

00:06:06.229 --> 00:06:07.688
we could write this as

00:06:07.688 --> 00:06:10.740
M to the one minus P,

00:06:10.740 --> 00:06:13.230
over one minus P

00:06:13.230 --> 00:06:16.852
minus, one to the one minus P,

00:06:16.852 --> 00:06:18.269
over one minus P.

00:06:19.454 --> 00:06:21.522
So now let's take the limits,

00:06:21.522 --> 00:06:23.945
so remember this integral,
we won't take the

00:06:23.945 --> 00:06:25.453
anti-derivative or the
definite integral here,

00:06:25.453 --> 00:06:26.758
but then we want to take the limit here as

00:06:26.758 --> 00:06:28.125
M approaches infinity.

00:06:28.125 --> 00:06:31.958
So what is the limit as
M approaches infinity,

00:06:32.863 --> 00:06:34.113
of natural log,

00:06:37.530 --> 00:06:38.551
of M?

00:06:38.551 --> 00:06:40.686
Well if M goes unbounded to infinity,

00:06:40.686 --> 00:06:42.527
well the natural log
of that is still going

00:06:42.527 --> 00:06:45.603
to go to infinity.

00:06:45.603 --> 00:06:48.436
So when P equals one, this
thing doesn't converge,

00:06:48.436 --> 00:06:50.936
this thing is just unbounded.

00:06:50.936 --> 00:06:55.103
So, P equals one, we diverge.

00:06:56.806 --> 00:06:58.072
So we know that.

00:06:58.072 --> 00:06:59.604
So now let's look over here,

00:06:59.604 --> 00:07:01.404
let's think about

00:07:01.404 --> 00:07:05.308
the limit as M approaches infinity,

00:07:05.308 --> 00:07:07.565
of this expression right over here.

00:07:07.565 --> 00:07:09.742
And the only part that's
really effected by

00:07:09.742 --> 00:07:11.602
the limit, is the part that has M,

00:07:11.602 --> 00:07:13.271
so we could even write
this as, we could take

00:07:13.271 --> 00:07:16.096
this one over one minus P out of this,

00:07:16.096 --> 00:07:19.212
we could say one over one minus P,

00:07:19.212 --> 00:07:20.924
times the limit as M approaches infinity,

00:07:20.924 --> 00:07:23.317
of M to the one minus P,

00:07:23.317 --> 00:07:25.564
and then separately we can subtract

00:07:25.564 --> 00:07:28.212
one to the one minus P, for any exponent

00:07:28.212 --> 00:07:30.247
that's just gonna be one,

00:07:30.247 --> 00:07:32.296
over one minus P.

00:07:32.296 --> 00:07:33.129
Is that right?

00:07:33.129 --> 00:07:34.977
Yeah no matter what exponent I put up here

00:07:34.977 --> 00:07:38.426
one to any power is going to be one.

00:07:38.426 --> 00:07:40.530
And so the interesting thing about whether

00:07:40.530 --> 00:07:44.116
it converges or not is,
this part of the expression.

00:07:44.116 --> 00:07:46.147
And it's all going to
depend on whether this

00:07:46.147 --> 00:07:49.252
exponent is positive or negative.

00:07:49.252 --> 00:07:53.252
If one minus P is greater than zero,

00:07:55.207 --> 00:07:56.925
if I'm going to infinity and I'm taking

00:07:56.925 --> 00:08:00.505
that thing to a positive exponent,

00:08:00.505 --> 00:08:02.353
then this is going to diverge.

00:08:02.353 --> 00:08:04.537
So in this situation we diverge, and

00:08:04.537 --> 00:08:05.817
one minus P is greater than zero we can

00:08:05.817 --> 00:08:08.331
add P to both sides, that's the situation,

00:08:08.331 --> 00:08:11.541
that's the same thing as one being greater

00:08:11.541 --> 00:08:14.995
than P, or P being less than one,

00:08:14.995 --> 00:08:17.690
we are going to diverge.

00:08:17.690 --> 00:08:19.304
So far we know that P is
going to be greater than

00:08:19.304 --> 00:08:22.154
zero, and we saw if P
is one, or if it's less

00:08:22.154 --> 00:08:24.253
than one, we're gonna diverge.

00:08:24.253 --> 00:08:26.678
But if this exponent right
over here is negative,

00:08:26.678 --> 00:08:30.258
if one minus P is less than zero,

00:08:30.258 --> 00:08:32.815
well think about it, then it's gonna be

00:08:32.815 --> 00:08:35.486
one over M to some positive exponent,

00:08:35.486 --> 00:08:37.088
is one way to think about it.

00:08:37.088 --> 00:08:38.428
So as M approaches infinity, this

00:08:38.428 --> 00:08:40.655
whole thing is going to approach zero.

00:08:40.655 --> 00:08:42.556
So this is actually
going to be a situation

00:08:42.556 --> 00:08:45.952
where we converge, where
we get to a finite value.

00:08:45.952 --> 00:08:47.375
And so we add P to both sides,

00:08:47.375 --> 00:08:51.522
we have one is less than P, we converge.

00:08:51.522 --> 00:08:53.053
So there you have it, we have established,

00:08:53.053 --> 00:08:56.066
this integral is going to converge only in

00:08:56.066 --> 00:08:58.819
the situation where P is greater than one.

00:08:58.819 --> 00:09:01.236
P>1 you're going to converge.

00:09:02.522 --> 00:09:07.287
And if zero is less than P
is less than or equal to one,

00:09:07.287 --> 00:09:09.035
you are going to diverge.

00:09:09.035 --> 00:09:10.702
And those are then the exact,

00:09:10.702 --> 00:09:12.883
cause this, our p-Series converges if

00:09:12.883 --> 00:09:15.345
and only if, this integral converges.

00:09:15.345 --> 00:09:17.810
And so these exact same constraints

00:09:17.810 --> 00:09:20.700
apply to our original p-Series.

00:09:20.700 --> 00:09:23.204
Our original p-Series converges

00:09:23.204 --> 00:09:24.627
only in the situation where

00:09:24.627 --> 00:09:28.628
P is greater than one, then we converge.

00:09:28.628 --> 00:09:32.324
And if zero is less than P
is less than or equal to one,

00:09:32.324 --> 00:09:35.574
we diverge, there you go.


Worked example: p-series

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.472
- [Instructor] So we have
an infinite series here,

00:00:02.472 --> 00:00:04.399
one plus one over two to the fifth,

00:00:04.399 --> 00:00:05.585
plus one over three to the fifth,

00:00:05.585 --> 00:00:07.595
and we just keep on going forever.

00:00:07.595 --> 00:00:09.953
We could write this as

00:00:09.953 --> 00:00:11.519
the sum

00:00:11.519 --> 00:00:12.936
from n equals one

00:00:14.795 --> 00:00:16.295
to infinity of one

00:00:17.728 --> 00:00:19.895
over n to the fifth power,

00:00:21.041 --> 00:00:23.497
one over n to the fifth power.

00:00:23.497 --> 00:00:26.831
And now you might recognize,
notice, when n is equal to one,

00:00:26.831 --> 00:00:28.204
this is one over one to the fifth,

00:00:28.204 --> 00:00:30.624
that's that over there,
and we could keep on going.

00:00:30.624 --> 00:00:34.441
Now you might immediately
recognize this as a p-series,

00:00:34.441 --> 00:00:38.481
and a p-series has the
general form of the sum,

00:00:38.481 --> 00:00:41.470
going from n equals one to infinity,

00:00:41.470 --> 00:00:42.553
of one over n

00:00:43.675 --> 00:00:47.273
to the p, where p is a positive value.

00:00:47.273 --> 00:00:48.899
So in this particular case,

00:00:48.899 --> 00:00:52.287
our p, for this p-series,
is equal to five.

00:00:52.287 --> 00:00:54.458
P is equal to five.

00:00:54.458 --> 00:00:58.110
Now you might already recognize,
under which conditions

00:00:58.110 --> 00:01:01.288
for a p-series does it
converge or diverge?

00:01:01.288 --> 00:01:03.676
It's going to converge.

00:01:03.676 --> 00:01:08.066
It's going to converge when
your p is greater than one,

00:01:08.066 --> 00:01:10.655
which is clearly the case in
this scenario right over here.

00:01:10.655 --> 00:01:13.691
Our p is clearly greater than one.

00:01:13.691 --> 00:01:15.652
We would diverge,

00:01:15.652 --> 00:01:16.985
we would diverge

00:01:18.197 --> 00:01:19.114
if our p is

00:01:20.288 --> 00:01:22.395
greater than zero

00:01:22.395 --> 00:01:25.009
and less than or equal,

00:01:25.009 --> 00:01:28.008
or less than or equal to one.

00:01:28.008 --> 00:01:29.166
This would be a divergent.

00:01:29.166 --> 00:01:31.748
So if this was like .9 here,

00:01:31.748 --> 00:01:34.065
or if this was a, you know, 3/4,

00:01:34.065 --> 00:01:36.118
then we would be diverging.

00:01:36.118 --> 00:01:39.977
So at least for this
one, we are convergent.

00:01:39.977 --> 00:01:43.013
Let's do another one of these.

00:01:43.013 --> 00:01:44.654
All right.

00:01:44.654 --> 00:01:49.156
So here, you might again
recognize this as a p-series.

00:01:49.156 --> 00:01:51.863
Let me rewrite this infinite sum.

00:01:51.863 --> 00:01:53.587
So this is the sum

00:01:53.587 --> 00:01:56.570
from n equals one to infinity

00:01:56.570 --> 00:01:57.570
of one over,

00:01:59.251 --> 00:02:01.048
let's see, we have a square root of two,

00:02:01.048 --> 00:02:02.120
a square root of three.

00:02:02.120 --> 00:02:03.518
So you could do this as two to the 1/2,

00:02:03.518 --> 00:02:05.433
three to the 1/2, four to the 1/2.

00:02:05.433 --> 00:02:08.459
So it's one over n to the 1/2.

00:02:08.459 --> 00:02:10.900
Notice, this is when n is equal to one.

00:02:10.900 --> 00:02:13.125
One over one to the 1/2 is one.

00:02:13.125 --> 00:02:16.056
One over two to the 1/2, well,
that's this right over here.

00:02:16.056 --> 00:02:18.407
And we keep on going on and on and on.

00:02:18.407 --> 00:02:20.511
Well, in this case, we
still have a p-series.

00:02:20.511 --> 00:02:22.313
We have one over n to some power,

00:02:22.313 --> 00:02:24.512
and that power is positive.

00:02:24.512 --> 00:02:28.837
But notice, in this
case, our p falls between

00:02:28.837 --> 00:02:30.703
zero and one.

00:02:30.703 --> 00:02:33.463
So 1/2 is our p.

00:02:33.463 --> 00:02:36.301
So p for our p-series is equal to 1/2,

00:02:36.301 --> 00:02:38.151
and that's between zero and one.

00:02:38.151 --> 00:02:40.323
Remember, we're divergent,

00:02:40.323 --> 00:02:41.156
divergent,

00:02:42.168 --> 00:02:45.528
when our p is greater than zero

00:02:45.528 --> 00:02:46.917
and less than or equal to one,

00:02:46.917 --> 00:02:49.160
which was clearly the
case right over here.

00:02:49.160 --> 00:02:51.910
So this is going to be divergent.


Partial sums: term value from partial sum

WEBVTT
Kind: captions
Language: en

00:00:00.163 --> 00:00:02.679
- [Instructor] We're told
that the nth partial sum

00:00:02.679 --> 00:00:05.207
of the series from n
equals one to infinity

00:00:05.207 --> 00:00:06.847
of a sub n is given by.

00:00:06.847 --> 00:00:09.743
And so the sum of the first
n terms is n squared plus one

00:00:09.743 --> 00:00:11.291
over n plus one.

00:00:11.291 --> 00:00:12.494
And they want us to figure out,

00:00:12.494 --> 00:00:14.637
what is the actual seventh term?

00:00:14.637 --> 00:00:16.808
And, like always, pause this
video and see if you can

00:00:16.808 --> 00:00:18.597
figure it out on your own before

00:00:18.597 --> 00:00:21.152
we work through it together.

00:00:21.152 --> 00:00:25.411
All right, so, one way to
think about it is: a sub seven,

00:00:25.411 --> 00:00:28.358
let's think about how that
relates to different sums.

00:00:28.358 --> 00:00:32.478
So, if we have a sub one,
plus a sub two, I'll just

00:00:32.478 --> 00:00:36.228
go all the way, a sub
three, plus a sub four,

00:00:37.707 --> 00:00:41.874
plus a sub five, plus a
sub six, plus a sub seven.

00:00:44.204 --> 00:00:47.274
So, if I were to sum all
of these things together,

00:00:47.274 --> 00:00:51.357
that--this entire sum--that
would be s sub seven.

00:00:53.038 --> 00:00:55.304
And if I wanted to figure out a sub seven,

00:00:55.304 --> 00:00:57.280
well I could subtract from that.

00:00:57.280 --> 00:01:00.900
I could subtract out the
sum of the first six terms.

00:01:00.900 --> 00:01:03.483
I could subtract out s sub six.

00:01:08.200 --> 00:01:11.140
So, once again, what am I doing
here, what is my strategy?

00:01:11.140 --> 00:01:14.593
I know the formula for the
sum of the first n terms.

00:01:14.593 --> 00:01:16.441
I can use that to say, okay,

00:01:16.441 --> 00:01:18.628
I can figure out the sum
of the first seven terms.

00:01:18.628 --> 00:01:20.260
That's gonna be the sum of all of these.

00:01:20.260 --> 00:01:21.869
And then I can use that
same formula to figure out

00:01:21.869 --> 00:01:24.216
the sum of the first six
terms, and the difference

00:01:24.216 --> 00:01:29.013
between the two, that's
going to be our a sub seven.

00:01:29.013 --> 00:01:32.177
So, another way of saying
what I just said is that

00:01:32.177 --> 00:01:36.344
a sub seven is going to be the
sum of the first seven terms,

00:01:37.358 --> 00:01:40.441
minus the sum of the first six terms.

00:01:43.848 --> 00:01:45.817
And if you were doing this
problem on your own you wouldn't

00:01:45.817 --> 00:01:47.937
have to write it out this way,
I just wrote it out this way,

00:01:47.937 --> 00:01:51.700
hopefully making this statement
a little bit more intuitive.

00:01:51.700 --> 00:01:54.690
Well, what is this going to be?

00:01:54.690 --> 00:01:58.302
Well, s sub seven, the sum
of the first seven terms,

00:01:58.302 --> 00:02:00.858
we just, wherever we see an
n, we replace it with a seven.

00:02:00.858 --> 00:02:04.839
So it's going to be
seven squared plus one,

00:02:04.839 --> 00:02:09.006
over seven plus one and from
that we are going to subtract

00:02:10.394 --> 00:02:13.243
s sub six, the sum of the first six terms.

00:02:13.243 --> 00:02:17.561
Well, that's going to
be six squared plus one,

00:02:17.561 --> 00:02:18.894
or six plus one.

00:02:21.414 --> 00:02:24.877
And from here we just have to
do a little bit of arithmetic.

00:02:24.877 --> 00:02:28.167
So, this is going to be
seven squared plus one,

00:02:28.167 --> 00:02:32.663
this is 49 plus one so
that is 50 over eight.

00:02:32.663 --> 00:02:35.413
And this is six squared plus one,

00:02:36.533 --> 00:02:37.366
that's 37,

00:02:38.444 --> 00:02:39.361
over seven.

00:02:40.423 --> 00:02:42.717
So, we want to find a common denominator

00:02:42.717 --> 00:02:44.922
between eight and seven.

00:02:44.922 --> 00:02:47.022
That would be 56.

00:02:47.022 --> 00:02:50.439
So this is going to be something over 56,

00:02:52.870 --> 00:02:55.287
minus something else over 56.

00:02:58.324 --> 00:03:01.677
Now, to go from eight to
56 I multiply by seven,

00:03:01.677 --> 00:03:04.210
so I need to multiple the
numerator by seven as well.

00:03:04.210 --> 00:03:06.043
50 times seven is 350.

00:03:08.565 --> 00:03:11.277
And then, this second fraction,
I multiple the denominator

00:03:11.277 --> 00:03:14.618
by eight to get to 56,
so I have to multiply

00:03:14.618 --> 00:03:15.868
37 times eight.

00:03:17.538 --> 00:03:19.996
Let's see 37 times eight is going to be

00:03:19.996 --> 00:03:22.329
240 plus 56, so that is 296.

00:03:27.051 --> 00:03:30.075
And so this is going to be equal to--

00:03:30.075 --> 00:03:34.242
so I have a denominator of
56, 350 minus 296 is 54.

00:03:36.648 --> 00:03:39.050
So, it's 54/56.

00:03:39.050 --> 00:03:41.545
And if we wanted to
reduce this a little bit,

00:03:41.545 --> 00:03:45.338
before we rewrite it
maybe in a simpler form,

00:03:45.338 --> 00:03:47.606
we're rewriting the same value.

00:03:47.606 --> 00:03:51.773
This would be, let's see, could
we write it as 27 over 28?

00:03:54.621 --> 00:03:55.497
Let's see, is that about?

00:03:55.497 --> 00:03:58.458
Yep, that's about as
simplified as we can get.

00:03:58.458 --> 00:04:02.896
There you go, that's what
a sub seven is, it's 27/28.

00:04:02.896 --> 00:04:04.949
The difference between the
sum of the first seven terms,

00:04:04.949 --> 00:04:07.866
and the sum of the first six terms.


Partial sums: formula for nth term from partial sum

WEBVTT
Kind: captions
Language: en

00:00:00.498 --> 00:00:02.147
- [Narrator] Nth partial
sum of the series,

00:00:02.147 --> 00:00:04.027
we're going from one
to infinity, summing it

00:00:04.027 --> 00:00:05.869
a sub n is given by.

00:00:05.869 --> 00:00:07.748
And they tell us of the formula

00:00:07.748 --> 00:00:09.197
for the sum of the first n terms.

00:00:09.197 --> 00:00:12.133
And they say write a rule for what

00:00:12.133 --> 00:00:15.726
the actual Nth term is going to be.

00:00:15.726 --> 00:00:17.322
Now to help us with this, let me just

00:00:17.322 --> 00:00:19.382
create a little visualization here.

00:00:19.382 --> 00:00:23.193
So if I have a sub one
plus, plus a sub two,

00:00:23.193 --> 00:00:26.110
plus a sub three, and I keep adding

00:00:27.048 --> 00:00:30.131
all the way to a sub n minus one plus

00:00:31.542 --> 00:00:32.375
a sub n.

00:00:34.038 --> 00:00:35.923
This whole thing, this whole thing that

00:00:35.923 --> 00:00:38.060
I just wrote out.

00:00:38.060 --> 00:00:39.979
That is s sub n.

00:00:39.979 --> 00:00:43.157
This whole thing is s, let me

00:00:43.157 --> 00:00:45.324
this whole thing is sub n,

00:00:46.237 --> 00:00:48.603
which is equal to n plus one,

00:00:48.603 --> 00:00:49.853
over n plus 10.

00:00:51.024 --> 00:00:52.761
Now if I wanna figure out a sub n,

00:00:52.761 --> 00:00:54.835
which is the goal of this exercise,

00:00:54.835 --> 00:00:56.906
well I could subtract out the sum of the

00:00:56.906 --> 00:00:59.688
first n minus one terms.

00:00:59.688 --> 00:01:02.105
So I could subtract out this.

00:01:03.896 --> 00:01:04.979
So that is s,

00:01:06.717 --> 00:01:08.884
that is s sub n minus one,

00:01:10.341 --> 00:01:11.711
and what would that be equal to.

00:01:11.711 --> 00:01:13.230
Well wherever we see
an n, we'd replace with

00:01:13.230 --> 00:01:14.314
an n minus one.

00:01:14.314 --> 00:01:17.091
So it'd be n minus one plus one over

00:01:17.091 --> 00:01:20.424
n minus one plus 10, which is equal to n

00:01:22.931 --> 00:01:24.348
over n plus nine.

00:01:25.789 --> 00:01:28.563
So if you subtract the red
stuff from the blue stuff,

00:01:28.563 --> 00:01:30.311
all that you're gonna be
left with is the thing

00:01:30.311 --> 00:01:31.421
that we're gonna solve for.

00:01:31.421 --> 00:01:33.417
You're gonna be left with a sub n.

00:01:33.417 --> 00:01:37.584
So we could write down a
sub n is equal to s sub n,

00:01:38.863 --> 00:01:42.446
is equal to s sub n
minus s sub n minus one

00:01:44.221 --> 00:01:46.043
S sub n minus one.

00:01:46.043 --> 00:01:49.960
Or we could write, that
is equal to this stuff.

00:01:50.984 --> 00:01:54.317
So this is the n plus one over n plus 10

00:01:56.229 --> 00:01:58.896
minus, minus n over n plus nine.

00:02:02.279 --> 00:02:05.175
And this by itself, this
is a rule for a sub n.

00:02:05.175 --> 00:02:06.583
But we could combine these terms,

00:02:06.583 --> 00:02:08.055
add these two fractions together.

00:02:08.055 --> 00:02:09.196
And this is actually going to be the case

00:02:09.196 --> 00:02:10.979
for n greater than one.

00:02:10.979 --> 00:02:13.809
For n equals one, s
sub one is going to be,

00:02:13.809 --> 00:02:17.184
well you can just, a sub
one is going to be equal

00:02:17.184 --> 00:02:18.600
to s sub one.

00:02:18.600 --> 00:02:20.908
But then for any other
n, we could use this

00:02:20.908 --> 00:02:21.971
right over here.

00:02:21.971 --> 00:02:24.152
And if we want to
simplify this, well we can

00:02:24.152 --> 00:02:25.927
add these two fractions.

00:02:25.927 --> 00:02:27.667
We can add these two fractions by having

00:02:27.667 --> 00:02:29.174
a common denominator.

00:02:29.174 --> 00:02:30.146
So let's see, if we multiply the

00:02:30.146 --> 00:02:32.093
numerator and denominator
here times n plus nine,

00:02:32.093 --> 00:02:33.760
we are going to get,

00:02:34.989 --> 00:02:38.822
this is equal to n plus
one times n plus nine,

00:02:40.414 --> 00:02:43.164
over n plus 10 times n plus nine.

00:02:46.334 --> 00:02:49.398
And from that, we are going to subtract.

00:02:49.398 --> 00:02:50.779
Let's multiply the
numerator and denominator

00:02:50.779 --> 00:02:52.671
here by n plus 10.

00:02:52.671 --> 00:02:55.088
So we have n times n plus 10,

00:02:56.297 --> 00:02:59.047
over n plus nine times n plus 10.

00:03:01.412 --> 00:03:03.745
N plus nine times n plus 10.

00:03:05.267 --> 00:03:07.423
And what does that give us?

00:03:07.423 --> 00:03:10.095
So, if we simplify up
here, we're gonna have,

00:03:10.095 --> 00:03:14.262
this is n squared plus 10
n plus nine, that's that.

00:03:18.254 --> 00:03:20.884
And then this right over
here this is n squared

00:03:20.884 --> 00:03:23.634
plus 10 n, within that red color.

00:03:27.232 --> 00:03:29.732
So this is n squared plus 10 n

00:03:31.536 --> 00:03:33.199
and remember we're gonna subtract this.

00:03:33.199 --> 00:03:36.105
And so, and we are close
to deserving a drum roll,

00:03:36.105 --> 00:03:39.779
a sub n is going to be
equal to our denominator

00:03:39.779 --> 00:03:43.696
right over here is n plus
nine times n plus 10.

00:03:45.266 --> 00:03:48.004
And we're gonna subtract the
red stuff from the blue stuff.

00:03:48.004 --> 00:03:50.188
So you subtract an n
squared from an n squared

00:03:50.188 --> 00:03:51.594
and those cancel out.

00:03:51.594 --> 00:03:54.111
Subtract a 10 n from a
10 n, those cancel out.

00:03:54.111 --> 00:03:56.772
And you're just left with that blue nine.

00:03:56.772 --> 00:03:58.587
So there you have it, we've expressed,

00:03:58.587 --> 00:04:02.754
we've written a rule for a
sub n, for n greater than one.


Overview of the Roman Empire

WEBVTT
Kind: captions
Language: en

00:00:00.227 --> 00:00:03.055
- [Instructor] When you
hear of ancient Rome,

00:00:03.055 --> 00:00:05.658
or the Roman Empire,
or the Roman republic,

00:00:05.658 --> 00:00:09.803
immediately images of the
Roman legions come to mind.

00:00:09.803 --> 00:00:12.276
To the conquering armies conquering

00:00:12.276 --> 00:00:13.716
much of the Mediterranean.

00:00:13.716 --> 00:00:16.087
You might have images of the Roman Senate.

00:00:16.087 --> 00:00:19.676
Names like Julius Caesar and
Augustus might come to mind.

00:00:19.676 --> 00:00:22.577
You might think of some of
the famous architectures.

00:00:22.577 --> 00:00:25.625
Some of which you can still
see if you were to visit Rome,

00:00:25.625 --> 00:00:28.744
and these are all real images of Rome,

00:00:28.744 --> 00:00:30.745
or they're appropriately
associated with it,

00:00:30.745 --> 00:00:33.617
and we'll talk about most of
these things in some depth,

00:00:33.617 --> 00:00:36.157
but Rome did not start out that way,

00:00:36.157 --> 00:00:37.901
and the purpose of this
video is to give us

00:00:37.901 --> 00:00:41.447
a over arching arc of the history of Rome

00:00:41.447 --> 00:00:43.847
to be able to place it within history.

00:00:43.847 --> 00:00:46.839
Both in terms of time and geography.

00:00:46.839 --> 00:00:49.886
So just to make sure we
can read what I did here

00:00:49.886 --> 00:00:53.743
ahead of time, up here is I
have a high level timeline

00:00:53.743 --> 00:00:56.759
and then down here I have
a timeline that zooms in

00:00:56.759 --> 00:00:58.977
a little bit, goes into
a few more details,

00:00:58.977 --> 00:01:01.428
and you can see this timeline on top,

00:01:01.428 --> 00:01:04.440
it's going from the eighth century BCE

00:01:04.440 --> 00:01:08.189
all the way to the fifth century CE,

00:01:08.189 --> 00:01:11.940
so it's covering over
1,000 years of history,

00:01:11.940 --> 00:01:14.788
and I need it to cover
over 1,000 years of history

00:01:14.788 --> 00:01:16.309
because the Roman Empire,

00:01:16.309 --> 00:01:19.165
we can even say just
the western Roman Empire

00:01:19.165 --> 00:01:20.469
covers that much,

00:01:20.469 --> 00:01:23.097
and that doesn't even
cover the entire legacy

00:01:23.097 --> 00:01:26.991
of the Roman Empire
because when we get into

00:01:26.991 --> 00:01:28.824
the fourth century CE,

00:01:30.955 --> 00:01:33.285
you have a split where
you have the eastern

00:01:33.285 --> 00:01:34.908
and the western Roman Empire

00:01:34.908 --> 00:01:37.741
and the eastern, often known
as the Byzantine Empire,

00:01:37.741 --> 00:01:39.991
that goes on until 1453 CE.

00:01:41.342 --> 00:01:45.015
So another 1,000 years
until they are conquered

00:01:45.015 --> 00:01:46.428
by the Ottomans.

00:01:46.428 --> 00:01:48.502
So the legacy of the Roman Empire,

00:01:48.502 --> 00:01:51.041
even formally as an
empire, is significant,

00:01:51.041 --> 00:01:52.966
and then it continues on.

00:01:52.966 --> 00:01:54.947
Much of western civilization,

00:01:54.947 --> 00:01:56.989
especially Europe and the Mediterranean

00:01:56.989 --> 00:01:59.529
has its foundations in the Roman Empire

00:01:59.529 --> 00:02:01.767
and then before that, Greek civilization,

00:02:01.767 --> 00:02:03.792
and the Roman Empire is really up there

00:02:03.792 --> 00:02:05.923
alongside the Persian Empire

00:02:05.923 --> 00:02:10.104
as one of the really great
civilizations or empires,

00:02:10.104 --> 00:02:11.832
and when I say great, you should take that

00:02:11.832 --> 00:02:12.922
with a grain of salt.

00:02:12.922 --> 00:02:15.151
Great I'm saying it was
big, it was powerful,

00:02:15.151 --> 00:02:16.874
but not everything they did was great.

00:02:16.874 --> 00:02:18.839
They had a lot of slavery,
they were very cruel,

00:02:18.839 --> 00:02:20.522
they were sometimes very violent.

00:02:20.522 --> 00:02:23.885
So take these terms great
with a grain of salt.

00:02:23.885 --> 00:02:25.973
So now that we get this timeline up here

00:02:25.973 --> 00:02:28.458
and then down here I'm gonna go, whoops.

00:02:28.458 --> 00:02:30.625
Down here I have some maps

00:02:31.777 --> 00:02:33.870
that are gonna show how
the Roman Empire grew

00:02:33.870 --> 00:02:36.201
and then eventually splits and declines,

00:02:36.201 --> 00:02:39.546
but as we go to this top timeline,

00:02:39.546 --> 00:02:43.713
we see the founding of Rome
in the eighth century BCE,

00:02:44.702 --> 00:02:48.143
and this date 753 is the
date that's often given

00:02:48.143 --> 00:02:51.047
to the founding of Rome by Romulus,

00:02:51.047 --> 00:02:55.968
one of twin brothers Romulus
and Remus raised by a she-wolf.

00:02:55.968 --> 00:02:58.032
They were abandoned and
raised by the she-wolf

00:02:58.032 --> 00:02:59.145
as legend would have it,

00:02:59.145 --> 00:03:01.027
and then Romulus eventually
kills his brother

00:03:01.027 --> 00:03:03.086
and then becomes the first king of Rome.

00:03:03.086 --> 00:03:05.274
We don't know how much of this is true.

00:03:05.274 --> 00:03:08.333
I suspect a lot of this is very legendary.

00:03:08.333 --> 00:03:12.252
Legend would have it that
Rome is named for Romulus,

00:03:12.252 --> 00:03:14.192
but some historians today think

00:03:14.192 --> 00:03:15.996
well it might have been
the other way around.

00:03:15.996 --> 00:03:18.043
We have the city of Rome,
they needed a founding story,

00:03:18.043 --> 00:03:20.770
hey let's say this person
Romulus started it,

00:03:20.770 --> 00:03:22.413
and we don't know the exact date,

00:03:22.413 --> 00:03:24.709
but there seems to be
a reasonable consensus

00:03:24.709 --> 00:03:28.731
that around this mid eighth century BCE

00:03:28.731 --> 00:03:31.228
you have the founding of the city of Rome,

00:03:31.228 --> 00:03:33.373
but it's important to
realize that at that point

00:03:33.373 --> 00:03:36.338
the founding of the city
that the kingdom of Rome,

00:03:36.338 --> 00:03:40.157
it wasn't a significant power
on the Italian Peninsula

00:03:40.157 --> 00:03:41.151
at the time.

00:03:41.151 --> 00:03:44.657
Where I have this X marked,
that's where Rome is,

00:03:44.657 --> 00:03:46.768
and what you see in
green, this is actually

00:03:46.768 --> 00:03:48.663
the Etruscan civilization,

00:03:48.663 --> 00:03:52.746
and for most of this period
of the Roman kingdom.

00:03:54.178 --> 00:03:56.274
So this period right over here,

00:03:56.274 --> 00:03:58.937
it's actually the Etruscans
that are dominant power

00:03:58.937 --> 00:04:01.111
and for significant fractions of this

00:04:01.111 --> 00:04:04.438
they were dominant over the Roman kingdom.

00:04:04.438 --> 00:04:06.152
Over the city of Rome,

00:04:06.152 --> 00:04:08.235
but as we get to 509 BCE,

00:04:09.784 --> 00:04:12.167
this is when the Roman
Republic is established

00:04:12.167 --> 00:04:14.522
and in this blue green
color, this is the kingdom.

00:04:14.522 --> 00:04:16.783
The gold color, this yellow
color is the republic

00:04:16.783 --> 00:04:20.236
and then the red color is when
the empire gets established

00:04:20.236 --> 00:04:23.859
by Julius Caesar and his
adopted son Augustus,

00:04:23.859 --> 00:04:26.127
and as we get into this,

00:04:26.127 --> 00:04:28.402
I guess you could say
this republic period of,

00:04:28.402 --> 00:04:29.934
or into the Roman Republic.

00:04:29.934 --> 00:04:32.184
This is when it starts
to really exert itself

00:04:32.184 --> 00:04:35.577
as more of a dominant
influence in the region,

00:04:35.577 --> 00:04:38.622
and it continues to be a
dominant influence in the region

00:04:38.622 --> 00:04:40.336
for several hundred years,

00:04:40.336 --> 00:04:44.894
and then as we get into the
first several hundred years

00:04:44.894 --> 00:04:46.969
in the common era.

00:04:46.969 --> 00:04:48.841
So after the time of Jesus.

00:04:48.841 --> 00:04:51.008
As we get into the 200s, the 300s,

00:04:51.008 --> 00:04:53.456
this is when we start
to see a real decline

00:04:53.456 --> 00:04:57.047
of the Roman Empire and
the western empire at least

00:04:57.047 --> 00:04:59.902
gets sacked multiple times
in the fifth century,

00:04:59.902 --> 00:05:03.079
and the year 476 is what's typically given

00:05:03.079 --> 00:05:05.827
for the end of the Roman Empire,

00:05:05.827 --> 00:05:08.793
but even that's not exactly
the end of the Roman Empire,

00:05:08.793 --> 00:05:10.526
because in the fourth century

00:05:10.526 --> 00:05:12.300
you have the Roman Empire splitting into

00:05:12.300 --> 00:05:14.183
this eastern and western empires,

00:05:14.183 --> 00:05:17.811
and the eastern later gets
known as the Byzantine Empire.

00:05:17.811 --> 00:05:20.811
That goes on for another 1,000 years

00:05:22.702 --> 00:05:25.115
until it's, as I
mentioned, I think earlier,

00:05:25.115 --> 00:05:26.865
taken over by the Ottomans,

00:05:26.865 --> 00:05:28.497
but to get more appreciation for

00:05:28.497 --> 00:05:31.284
how the Roman Empire grew

00:05:31.284 --> 00:05:33.138
and then eventually starts to shrink,

00:05:33.138 --> 00:05:35.997
I have some maps over here.

00:05:35.997 --> 00:05:39.724
So this first map, this
is roughly from the

00:05:39.724 --> 00:05:42.974
third century BCE and third century BCE

00:05:44.385 --> 00:05:47.262
you might remember Rome is
a republic at this time,

00:05:47.262 --> 00:05:51.417
but it's not the dominant
power in the Mediterranean.

00:05:51.417 --> 00:05:53.665
Just to give ourselves some bearings,

00:05:53.665 --> 00:05:58.283
you might remember that in
the later fourth century BCE

00:05:58.283 --> 00:06:01.586
that's when Alexander conquers much of,

00:06:01.586 --> 00:06:05.289
or most of Greece, Macedonia,
the Persian Empire.

00:06:05.289 --> 00:06:08.519
Which included at the time Egypt,

00:06:08.519 --> 00:06:11.260
and as we get into the third century BCE,

00:06:11.260 --> 00:06:15.154
you still have the fragments
of Alexanders empire,

00:06:15.154 --> 00:06:18.094
but these are still significant
powers in the regions.

00:06:18.094 --> 00:06:20.546
You also have the Carthaginian Empire.

00:06:20.546 --> 00:06:22.644
Which you see here in blue in North Africa

00:06:22.644 --> 00:06:24.269
and southern Spain,

00:06:24.269 --> 00:06:26.975
and this period you have a series of wars

00:06:26.975 --> 00:06:29.734
between the Romans and the Carthaginians

00:06:29.734 --> 00:06:31.166
known as the Punic Wars,

00:06:31.166 --> 00:06:33.534
and we'll do videos in
depth on the Punic Wars.

00:06:33.534 --> 00:06:35.072
They're called the Punic Wars

00:06:35.072 --> 00:06:37.086
because the term Punic, it comes from

00:06:37.086 --> 00:06:39.017
what the Romans called the Phoenicians,

00:06:39.017 --> 00:06:41.911
and Carthage was actually settled,

00:06:41.911 --> 00:06:43.668
founded by ancient Phoenicians.

00:06:43.668 --> 00:06:46.277
So you can kinda consider
this the Phoenician War

00:06:46.277 --> 00:06:48.925
or the descendants of the Phoenician Wars,

00:06:48.925 --> 00:06:51.827
but that's why it's called Punic,

00:06:51.827 --> 00:06:55.628
but the Romans eventually
win the multiple rounds

00:06:55.628 --> 00:06:57.837
of Punic Wars, take over their territory,

00:06:57.837 --> 00:07:00.234
and eventually destroy Carthage.

00:07:00.234 --> 00:07:02.984
Carthage is destroyed in 146 BCE,

00:07:04.016 --> 00:07:07.375
and we'll talk about that in more detail.

00:07:07.375 --> 00:07:11.369
Now as we go into the first century BCE,

00:07:11.369 --> 00:07:16.175
this is when Rome goes form
being a republic to an empire,

00:07:16.175 --> 00:07:19.998
and it happens when the
general, Julius Caesar,

00:07:19.998 --> 00:07:24.614
he's able to conquer Gaul
for the Roman Republic.

00:07:24.614 --> 00:07:26.947
So Gaul is modern day France

00:07:28.048 --> 00:07:30.554
and some other surrounding regions,

00:07:30.554 --> 00:07:35.429
and he is so powerful that
when he comes back to Rome,

00:07:35.429 --> 00:07:39.410
a civil war ensues and
Julius Caesar is able to

00:07:39.410 --> 00:07:41.225
essentially win the civil war,

00:07:41.225 --> 00:07:43.459
declare himself a dictator,

00:07:43.459 --> 00:07:47.338
and then his adopted son
Augustus, right over here,

00:07:47.338 --> 00:07:49.838
is in 27 BCE declared Emperor,

00:07:51.992 --> 00:07:55.341
and that's the beginning
of the Roman Empire,

00:07:55.341 --> 00:07:57.685
and we keep going, and I
think I've already mentioned

00:07:57.685 --> 00:07:59.036
it once in this video,

00:07:59.036 --> 00:08:03.953
but eventually we get a split
in the fourth century CE

00:08:03.953 --> 00:08:06.137
where we have eastern and western halves

00:08:06.137 --> 00:08:07.566
of the Roman Empire.

00:08:07.566 --> 00:08:10.621
The western side centered at Rome,

00:08:10.621 --> 00:08:14.756
the eastern side centered at
what was originally Byzantium,

00:08:14.756 --> 00:08:18.609
renamed Constantinople,
and today Istanbul,

00:08:18.609 --> 00:08:20.831
and as we get into the fifth century,

00:08:20.831 --> 00:08:22.357
that's when the western empire

00:08:22.357 --> 00:08:24.362
especially starts to
really go into decline

00:08:24.362 --> 00:08:27.270
and is eventually sacked in 476,

00:08:27.270 --> 00:08:30.437
but the eastern empire keeps on going.


Zoroastrianism

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.447
- [Instructor] So in any
discussion of ancient Persia,

00:00:02.447 --> 00:00:06.245
we quickly talk about the
faith of the Achaemenid Empire,

00:00:06.245 --> 00:00:07.842
and that's Zoroastrianism,

00:00:07.842 --> 00:00:09.946
and it's popularized by Cyrus the Great

00:00:09.946 --> 00:00:12.815
when he establishes the Achaemenid Empire,

00:00:12.815 --> 00:00:14.251
takes over the Median Empire,

00:00:14.251 --> 00:00:17.473
the Neo-Babylonian Empire, the Lydians.

00:00:17.473 --> 00:00:20.216
His son, Cambyses, takes
over the Egyptians.

00:00:20.216 --> 00:00:23.895
And their faith is based on
the teaching of Zoroaster.

00:00:23.895 --> 00:00:27.222
And Zoroaster is the Greek
name for Zarathustra,

00:00:27.222 --> 00:00:29.102
and sometimes the religion is referred to

00:00:29.102 --> 00:00:31.145
as Zarathustrianism.

00:00:31.145 --> 00:00:33.844
And it's one of the
oldest religions we have,

00:00:33.844 --> 00:00:35.175
and it's an existing religion.

00:00:35.175 --> 00:00:38.282
People still practice Zoroastrianism.

00:00:38.282 --> 00:00:39.992
In terms of when it started,

00:00:39.992 --> 00:00:41.929
there's some debate.

00:00:41.929 --> 00:00:43.590
A lot of accounts have it starting

00:00:43.590 --> 00:00:46.670
as early or even earlier
than 3,500 years ago,

00:00:46.670 --> 00:00:48.739
so around 1500 BCE,

00:00:48.739 --> 00:00:50.449
and some accounts put
it a little bit closer

00:00:50.449 --> 00:00:51.514
to Cyrus the Great,

00:00:51.514 --> 00:00:53.264
maybe 700 or 600 BCE.

00:00:54.829 --> 00:00:55.986
And the accounts,

00:00:55.986 --> 00:00:57.185
or the beginning of the religion

00:00:57.185 --> 00:01:01.447
is really around the
teachings of Zarathustra,

00:01:01.447 --> 00:01:06.142
and his teachings really
establish one of the first

00:01:06.142 --> 00:01:08.815
often viewed as monotheistic
religions of the world,

00:01:08.815 --> 00:01:11.250
this idea that there is one god,

00:01:11.250 --> 00:01:16.097
and that god in the Zoroastrian
faith is Ahura Mazda.

00:01:16.097 --> 00:01:17.718
Ahura Mazda.

00:01:17.718 --> 00:01:20.838
And it's viewed that Ahura Mazda

00:01:20.838 --> 00:01:22.714
does not take a human form.

00:01:22.714 --> 00:01:25.479
It is even, you cannot even conceive of,

00:01:25.479 --> 00:01:27.307
you know, fully conceive,

00:01:27.307 --> 00:01:30.839
the human mind can't fully
conceive of Ahura Mazda,

00:01:30.839 --> 00:01:32.756
but Ahura Mazda is the,

00:01:33.605 --> 00:01:34.750
in the Zoroastrian faith,

00:01:34.750 --> 00:01:36.515
the one and only god.

00:01:36.515 --> 00:01:40.655
But in conjunction with
the idea of Ahura Mazda,

00:01:40.655 --> 00:01:43.301
a lot of people refer to Zoroastrianism

00:01:43.301 --> 00:01:45.051
as a dualistic faith.

00:01:46.709 --> 00:01:49.618
A dualistic faith means that
there's this kind of interest

00:01:49.618 --> 00:01:51.868
in counterbalancing forces.

00:01:53.158 --> 00:01:55.741
For example, light versus dark,

00:01:57.917 --> 00:02:00.500
and of course good versus evil.

00:02:03.002 --> 00:02:06.379
And one of the main ideas or words here

00:02:06.379 --> 00:02:08.962
are the ideas of asha and druj.

00:02:10.472 --> 00:02:14.314
So asha, these are the
ideas of being principled,

00:02:14.314 --> 00:02:15.809
honest, ordered,

00:02:15.809 --> 00:02:17.892
and druj is the opposite.

00:02:18.824 --> 00:02:21.405
Druj is the idea of being unprincipled,

00:02:21.405 --> 00:02:23.403
dishonest, chaotic.

00:02:23.403 --> 00:02:25.494
And so there's a lot of talk

00:02:25.494 --> 00:02:29.661
and kind of insight or trying
to get a better understanding

00:02:30.988 --> 00:02:33.267
of these dualistic natures in the world,

00:02:33.267 --> 00:02:36.016
and trying for humanity be
on the side of the good,

00:02:36.016 --> 00:02:37.401
the light, the ordered,

00:02:37.401 --> 00:02:39.553
the principled, the honest.

00:02:39.553 --> 00:02:43.121
Now right over here is a
key symbol of Zoroastrianism

00:02:43.121 --> 00:02:43.954
that you might see.

00:02:43.954 --> 00:02:45.822
It's called the Faravahar.

00:02:45.822 --> 00:02:48.225
I'm probably be mispronouncing it.

00:02:48.225 --> 00:02:52.510
Faravahar, and there's
a lot of symbolism here.

00:02:52.510 --> 00:02:55.928
It shows the connection of
the universe to humanity,

00:02:55.928 --> 00:02:58.007
you see the depiction
of a human being here,

00:02:58.007 --> 00:03:00.334
and the three layers of the wings

00:03:00.334 --> 00:03:03.737
depict these three very important
ideas in Zoroastrianism.

00:03:03.737 --> 00:03:06.487
The first layer is good thoughts,

00:03:10.640 --> 00:03:14.807
the second layer right over
here of wings are good words,

00:03:15.892 --> 00:03:19.051
and then the third layer
here is good actions,

00:03:19.051 --> 00:03:20.362
good actions.

00:03:20.362 --> 00:03:21.195
And it makes sense.

00:03:21.195 --> 00:03:22.314
You kind of imagine this idea

00:03:22.314 --> 00:03:24.712
that good thoughts lead to good words,

00:03:24.712 --> 00:03:25.727
and that combined,

00:03:25.727 --> 00:03:27.245
they can lead to good actions.

00:03:27.245 --> 00:03:28.304
And on the tail here,

00:03:28.304 --> 00:03:29.903
and these are the things you should avoid,

00:03:29.903 --> 00:03:31.846
these are the bad thoughts,

00:03:31.846 --> 00:03:33.960
the bad actions,

00:03:33.960 --> 00:03:36.684
or the bad thoughts, the bad words,

00:03:36.684 --> 00:03:38.351
and the bad actions.

00:03:40.260 --> 00:03:42.462
Along lines this ideas of dualism,

00:03:42.462 --> 00:03:46.234
especially the fascination
with the ideas of light,

00:03:46.234 --> 00:03:49.703
fire is a very important element
of the Zoroastrian faith.

00:03:49.703 --> 00:03:51.527
In fact, if you visit
a Zoroastrian temple,

00:03:51.527 --> 00:03:53.584
you will see an eternal fire,

00:03:53.584 --> 00:03:56.223
and they feed that fire five times a day.

00:03:56.223 --> 00:04:00.345
Another very important
element of the faith is water.

00:04:00.345 --> 00:04:02.901
And the sacred text for the Zoroastrians

00:04:02.901 --> 00:04:04.651
is called the Avesta.

00:04:05.574 --> 00:04:07.936
And the Avesta is written in
a language called Avestan,

00:04:07.936 --> 00:04:09.598
and it's called Avestan
because it's the language

00:04:09.598 --> 00:04:11.132
that the Avesta was written in,

00:04:11.132 --> 00:04:13.609
but it's written in a language

00:04:13.609 --> 00:04:16.807
that is closely related
to ancient Sanskrit,

00:04:16.807 --> 00:04:19.537
and so there's probably some connections,

00:04:19.537 --> 00:04:21.453
or for sure there's some connections,

00:04:21.453 --> 00:04:25.399
between ancient Hinduism
and the progenitor

00:04:25.399 --> 00:04:27.562
or some of the preexisting ideas

00:04:27.562 --> 00:04:30.520
that might, that Zarathustra developed

00:04:30.520 --> 00:04:33.649
or might have built on top of.

00:04:33.649 --> 00:04:35.909
Now one of the really interesting things

00:04:35.909 --> 00:04:38.774
about Zoroastrianism is
that even though today

00:04:38.774 --> 00:04:41.188
it's a relatively small faith,

00:04:41.188 --> 00:04:43.147
on the order of less than 200,000 people

00:04:43.147 --> 00:04:44.235
around the world practice it,

00:04:44.235 --> 00:04:47.535
and most of them are in
India and the United States,

00:04:47.535 --> 00:04:49.033
at one time, it was,

00:04:49.033 --> 00:04:51.803
you could view it as the state
religion of ancient Persia,

00:04:51.803 --> 00:04:52.636
but then eventually,

00:04:52.636 --> 00:04:54.803
in the seventh century CE,

00:04:55.974 --> 00:04:59.057
when Persia, or Iran, became Islamic,

00:04:59.990 --> 00:05:02.684
you have a Zoroastrian diaspora,

00:05:02.684 --> 00:05:04.818
especially moving to
India and then eventually

00:05:04.818 --> 00:05:05.663
places like the United States,

00:05:05.663 --> 00:05:09.477
although there are still
practicing Zoroastrians in Iran.

00:05:09.477 --> 00:05:11.552
But above and beyond the continuing legacy

00:05:11.552 --> 00:05:13.717
of the Zoroastrians,

00:05:13.717 --> 00:05:15.192
they've had a lot of influence

00:05:15.192 --> 00:05:17.912
on some of the major
religions of the world.

00:05:17.912 --> 00:05:19.327
And what you have on this timeline here,

00:05:19.327 --> 00:05:20.993
I've made some of the major religions.

00:05:20.993 --> 00:05:24.066
You see up here in blue you have Judaism,

00:05:24.066 --> 00:05:25.771
and the reason why I
draw the dotted line is

00:05:25.771 --> 00:05:27.394
these are periods and it's not exact,

00:05:27.394 --> 00:05:29.395
you know, these religions are evolving

00:05:29.395 --> 00:05:31.006
and they are forming and they're getting

00:05:31.006 --> 00:05:32.764
more and more tradition over time.

00:05:32.764 --> 00:05:35.116
So you hear Judaism, and of course,

00:05:35.116 --> 00:05:38.677
Christianity is in a lot
ways built on top of Judaism.

00:05:38.677 --> 00:05:40.094
Jesus was Jewish.

00:05:41.287 --> 00:05:42.963
You have Islam right over here,

00:05:42.963 --> 00:05:46.076
which of the major world
religions is one of the newest,

00:05:46.076 --> 00:05:49.237
in a lot of ways built or related

00:05:49.237 --> 00:05:51.570
to Christianity and Judaism.

00:05:52.450 --> 00:05:56.057
And even though the Zoroastrianism

00:05:56.057 --> 00:06:00.058
is related and it kind
of comes out of a similar

00:06:00.058 --> 00:06:03.725
kind of early Aryan
people type of religion,

00:06:04.823 --> 00:06:06.391
it has a lot of influence

00:06:06.391 --> 00:06:09.670
on some of these other traditional,

00:06:09.670 --> 00:06:11.290
or these mainstream religions

00:06:11.290 --> 00:06:12.933
that we have in the world today.

00:06:12.933 --> 00:06:14.068
As we mentioned,

00:06:14.068 --> 00:06:17.211
it's viewed as a monotheistic religion,

00:06:17.211 --> 00:06:19.846
and it's also had direct influence on,

00:06:19.846 --> 00:06:21.897
for example, Christianity.

00:06:21.897 --> 00:06:23.949
So right over here,

00:06:23.949 --> 00:06:26.335
this is from John Bowker,

00:06:26.335 --> 00:06:29.166
who's a famous Christian theologian.

00:06:29.166 --> 00:06:30.538
He wrote in his book World Religions

00:06:30.538 --> 00:06:32.452
"that angels, the end of the world,

00:06:32.452 --> 00:06:34.366
"a final judgement, the resurrection,

00:06:34.366 --> 00:06:37.394
"and heaven and hell
received form and substance

00:06:37.394 --> 00:06:40.093
"from Zoroastrian beliefs."

00:06:40.093 --> 00:06:42.505
In Christian tradition,

00:06:42.505 --> 00:06:44.793
the Three Wise Men, the Three Kings,

00:06:44.793 --> 00:06:48.210
the Three Magi that visit the baby Jesus,

00:06:49.205 --> 00:06:52.562
Magi, this is referring
to Zoroastrian priests

00:06:52.562 --> 00:06:54.203
who were visiting the baby Jesus.

00:06:54.203 --> 00:06:57.079
And the word magic comes
from the word magi,

00:06:57.079 --> 00:07:00.717
because the Zoroastrian
elite, the priests,

00:07:00.717 --> 00:07:04.884
they were very capable in
the sciences of astronomy,

00:07:06.156 --> 00:07:08.612
and they were also very
well-known for their astrology,

00:07:08.612 --> 00:07:11.504
and so some people associated them with,

00:07:11.504 --> 00:07:14.229
you know, magical capabilities.

00:07:14.229 --> 00:07:17.017
It also has close ties to Islam.

00:07:17.017 --> 00:07:20.414
The sacred fire in Zoroastrianism
is fed five times a day.

00:07:20.414 --> 00:07:23.331
Zoroastrians pray five times a day,

00:07:24.223 --> 00:07:25.757
and before prayer,

00:07:25.757 --> 00:07:30.346
they go through a ritual
around washing themselves

00:07:30.346 --> 00:07:31.375
to cleanse themselves,

00:07:31.375 --> 00:07:34.123
where they wash themselves
three times before each prayer.

00:07:34.123 --> 00:07:36.884
That is something that we see in Islam.

00:07:36.884 --> 00:07:37.970
So Zoroastrianism,

00:07:37.970 --> 00:07:40.611
it's the faith of ancient Persia

00:07:40.611 --> 00:07:43.836
popularized by Cyrus the Great,

00:07:43.836 --> 00:07:46.933
and it's a living religion, even today,

00:07:46.933 --> 00:07:49.685
although it's a much smaller
population that practices it.

00:07:49.685 --> 00:07:50.634
But as we've seen,

00:07:50.634 --> 00:07:51.908
it's had a lot of influence

00:07:51.908 --> 00:07:54.991
on many of the major world religions.