Hi everyone. Welcome back to integralcalc.com.
We’re going to be doing a bunch of infinite
series problems today. The first one is one
plus e to the negative one plus e to the negative
two plus e to the negative three plus dot
dot dot.
It’s the series that we're given. The problem
asked us to determine whether or not this
infinite series converges or diverges and
if it does converge, to find the sum of the
series. So given that, if that's what the
question asks us to do, the formula that we
can use for that kind of problem is the following.
This notation is called summation notation,
this weird e thing is the, symbolizes the
sum. It just means the sum of this series
here when the first value of k is zero and
going on into infinity.
So the formula nonetheless is, it has three
parts, this is the first part. Second part,
this formula is much longer than this but
I just abbreviate it with these three parts
because they're really the only things you
need to remember. And then one over one minus
x. So this is the first part, the second part
and the third part. I’ve separated them
with a semi colon which I think I'm going
to take a way because it's kind of harder
to see.
So this is the formula. I’ve already explained
the first part but basically if your series
fits into this form meaning if you could apply
this x to the k to your series. If your series
takes this form, then you have your x right
here. If the absolute value of x is less than
one then your series converges and it converges
to this value, x being the same here as it
is in both of these. If it's greater than
or equal to one, the absolute value of x is
greater than or equal to one, then you have
a divergent series and you don't even need
to worry about this third step.
So basically what we need to do given that
this is a three part formula is determine
first whether or not our series takes this
form. If it does then determine whether or
not the absolute value of x is less than one,
if it does, we know it converges and then
we would need to proceed to this third step
to find the sum of the series. So three parts,
like I said. First thing, does our series
fit this form here? the way that, the easiest
way that I find to determine this, if your
series already begins with one, meaning the
first term is one like we have in our series
here. We have one and then plus a series of
e's and exponents, if the first term is one,
I don't know maybe ninety percent of the time,
a lot of the time, this term right here is
going to be your, is going to be x, is exactly
equal to x and you're one and you don't need
to go any farther. We’ll show you how you
can prove this but this; it's usually the
second term and that includes the sign here.
You know, if it's a positive sign, it doesn't
matter so much but if this were a negative
sign x would be negative e to the negative
one and you need to include that sign. So
since the series starts with one, I'm gonna
guess that this whole term, e to the negative
one is our x term.
The way that we can double check to make sure
that it is, is the following. Remember how
I said that this notation means that the first
value of k will be zero? Well, you can imagine
if the first value of your k is zero and we're
going to infinity, then this series looks
like the following. x to the zero plus x to
the one plus x to the two plus x to the three
plus dot dot dot forever. x to the zero, (right?)
is one because anything raised to the zero
power is one. So that's kind of why there's
one here in the beginning of the series. It’s
sort of a dead give away because, and if your
series, if the first value of k is zero and
this is a one, then anything to the zero power
and that begins the first term in the series
is equal to one so you know you're kind of
already set up. But let's go ahead and double
check. Now, and realize too that x to the
one is the same thing as x, which is why this
second term here, these are going to correspond
here. This one, the second term, this x and
what we're going to take here, this third
term, x squared, and the third term we're
going to get here so x to the one is the same
thing as x which is why this second term here
right after the one is our x term.
But let's go ahead and double check this anyway.
So this would be (right?), we said e to the
negative one was our x, so we have e to the
negative one and then to the one plus x again
is e to the negative one to the two plus again
e to the negative one, this time to the three
plus dot dot dot. Okay, so if our x term is
e to the negative one, then this is what our
series looks like and does this series match
this series? Well, if we simplify, we can
see that it does infact because we get e to
the negative one, when you have one exponent
raised to another exponent, all you do is
multiply them together. So since this is negative
one squared, that's simply the two multiplied
together and you get negative two, so e to
the negative two plus e, negative one times
three is negative three plus dot dot dot.
And you can see that this series down here
is matching up exactly to this series which
means that we did infact pick the correct
x, value of x here.
We did that correctly and our x value is e
to the negative one. So now that we've determined
our x value, that's really all we need to
do. We need to figure out what our series
took this form. We need to figure out what
x was since one was the first term in the
series, I predicted that this second term 
here would be our x term because it almsot
is when a series starts with one and then
I just double checked. This is the series,
the form that the series takes based on this
formula, so I plugged our x term into this
form here, simplify and yes they do match
which means that our series does take this
form and our x term is e to the negative one.
So since x e to the negative one, our series
will look like this. Infinity, k equals zero
and we'll have e to the negative one to the
k.
So that is our series. Now we need to determine
whether or not the series converges or diverges.
So what we go ahead and do is the absolute
value of our x which is e to the negative
one and now we do that on our calculator.
So when you're plugging this into your calculator,
make sure that you plug it in like this. e
and then parenthesis to the negative one because
and this of course you'll have a little carat
like that. So when I plug in on my calculator,
negative one, I get point three six seven
eight. If I, let's see if we we do something,
e to the negative one, yeah so it does end
up being the same thing. You don't technically
need the parenthesis but I think it's a good
practice to have since you have the negative
sign in there, you just have to make sure
that it's included in the right way but anyway,
this is the value you get when you plug in
e to the negative one.
So we're looking at the absolute value of
point three six seven eight. Absolute value
basically just means take the positive value
of whatever's inside these vertical brackets
here. So if this had been negative two, absolute
value of negative two is two because we just
make it positive even if it’s negtive. Since
our value is already positive, then absolute
value doesn't really apply. This ends up being
the same thing. It's just point three six
seven eight. Since point three six seven eight
is less than one, that does mean that our
series converges. Remember that if this value
is greater than or equal to one, then our
series diverges.
So we've determined that the series does converge
and they asked us to determine the sum if
the series did converge. So let's go ahead
and use the third part of our equation here
to determine the sum. So remember that x here,
we're using e to the negative one. So our
sum would be one over one minus e to the negative
one. So I'm going to go ahead and do same
simplification here. This would be the same
as one over one minus, since this is e to
the negative one, that's the same thing as
one over e. You can, this e to the negative
one here is in the numerator, technically
of its own term. We moved it to the denominator
by putting one over it, which means we can
change this negative one to a positive one
right here. Of course we didn't write it because
it's redundant. We never write anything to
the power of one. So this one over e is the
same thing as e to the negative one. So that's
some simplification. Then I'm, gonna go ahead
and change this one to e over e, right? Which
is the same thing as one so that I can combine
these two fractions on the bottom. So what
we'll end up having is one over and since
I have e in the denominators of both of these
fractions, I can combine them and say we minus
one over e, is my denominator here.
Now, since I have one divided by a fraction,
I can flip this fraction over on itself and
multiply these together. So instead of this
divided by this, I can do this multiplied
by the inverse of this fraction here. So this
is the same thing as one times e over e minus
one, which mean that that one can just go
away obviously.
And we're left with e over e minus one so
this is the cleanest way to leave our final
answer but if you want to convert to a decimal
or if you have multiple choice exam that maybe
only gives you decimals, then you would of
course do, I would do the bottom first. I
would do secant e to the one, subtract one
from that and then do secant e to the one
divided by answer and the decimal value of
that ends up being one point five eight one
nine.
So either of these, they're equivalent obviously
are correct, but this is the cleaner way to
leave your answer, so once you get here, I
would go ahead and stop. So again, find out
if the series is in this form, determine what
our x is, apply this formula to see whether
or not it converges or diverges, convergent
being when the  absolute value of x is less
than one, and then since it does converge,
find its sum using this formula here, just
a  quick algebraic simplification and that's
your final answer. Thanks, guys.
