PROFESSOR: Welcome back.
In our last exciting
episode, we were
talking about
alternating series,
which are series where
one term is positive,
then one term is negative,
then positive, then negative.
The signs alternate.
Those are actually pretty easy
to check if they converge.
Right?
We just see if the
numbers that we're adding
are subtracting, or
decreasing, and going to 0.
That's much easier to check and
it's also much easier for them
to converge.
For example, if we
look at the examples
we had last time,
maybe this first one,
often called the
alternating harmonic series
because it's the
sum of 1 over k.
By itself, that'd be
the harmonic series.
With the negative 1 there,
the negative 1 to k,
it makes the sign
switch back and forth.
So it's an alternating series.
And in this case,
because the 1 over k,
we call it the alternating
harmonic series.
If the alternator wasn't there,
we just have the harmonic
series and it would diverged.
But if we put it
in the negative 1
to the k to make the sign
switch back and forth,
it will converge by our
alternating series test.
So we need a way to describe
the difference between series
like this one that converge
because it has help,
because half the
terms are negative,
so we're subtracting as
much as we're adding.
So that's one possibility
of series converging,
but it is kind of a faker.
Right?
It's converging
because it has help,
because half the
terms are negative.
So if we had a series
like negative 1 of the k
and then 1 over k squared,
p series with p equal to 2,
that would converge even
without the negative 1 to the k.
So we need a way to
tell the two apart
and so our first
definition will do that.
A series will
converge absolutely
if it converges when you
make all the terms positive.
So the reason we say
converges absolutely ,
is not that we're
really certain about it.
It absolutely it converges.
What we mean is it converges if
we make all the terms positive,
the way you write that is to
put absolute values around all
the terms.
Right?
That's how you
would do it right,
make all the terms
positive is you put
absolute values around the ak.
So a series ak will
converge absolutely
if the sum with absolute
values around it
converges, absolute
values around each term.
So that's great, right?
So that gives us a way to say
if it converges absolutely,
then it really, really,
really does converge,
even if all the
terms were positive.
So those are sort
of the real deal.
Right?
What about the
fakie ones, right?
Well, if we have a series that
converges, but not absolutely.
So a great example is that
alternating harmonic series.
It converged because
it was alternating,
but without the alternator, if
we meet all the terms positive,
we just have the harmonic
series and it wouldn't converge.
If we have a series that
converges, but not absolutely,
again like maybe a
harmonic series, then
we say it converges
conditionally.
Again, the words aren't
necessarily great.
I mean, converges
absolutely makes
it sound like we
weren't sure about all
the other ones converging.
Right?
But converging absolutely has
to do with absolute values.
Converging conditionally, why
do you call it conditionally?
Well, it's sort of that
the condition is that half
the terms be negative.
That's what the
conditionally is,
it's converging
because of the help
it's getting from half
its terms being negative.
Well I say half,
these definitions
apply even if we're not talking
about a harmonic series.
And we'll see some of
these in the homework
where we just have a
series with a whole bunch
of negative terms,
infinitely many of the terms
are negative, infinitely
many of them are positive.
They don't have to alternate for
this definition to make sense.
OK let's do some examples.
So here is negative 1 to
the k and 1 over k squared.
Well, if we try the
alternating series test,
which would be the
obvious thing to try,
where we put our thumb over the
sum and over the alternator.
And we just look at the
sequence 1 over k squared.
The alternating series test
says we just have to check
is that decreasing?
Yeah.
Is it going to 0?
Duh, so this is clearly
going to converge
by the alternating series test.
But does it converge
absolutely or not?
Well that would be like
just putting your thumb
over the alternator.
So if I ignore this
part and I look
at the sum of the
absolute value of 1 over k
squared, in other words
ignoring the minuses.
So I have the sum
of 1 over k square
by itself, that would be a
p series, with p equal to 2.
That guy converges so this
series converges absolutely.
It converges even if I
put absolutely values
around the whole thing.
Next step, well
this is one we've
been talking about already,
alternating harmonic series.
So the some of negative 1
of the k and then 1 over k.
Again, this will
converge because if I
look at the 1 over k sequence
by itself, that's decreasing,
goes to 0, alternating series
test says it converges.
But if I take out
the alternator,
I've got the sum of just 1 over
k, that's the harmonic series
and it diverges.
So what I have left
converges but not absolutely.
So it converges conditionally.
So let's think a little bit
more carefully about the steps
we went through when
we worked this problem.
If we looked back at sum
of negative 1 to the k,
and then 1 over k squared.
So I thought to myself the
following thoughts, oh wow,
that looks like an
alternating series.
Yea it's easy.
Alternating series test
is what I need to do.
How do I do the
alternating series test?
I completely ignore the sum
and the negative 1 to the k,
and I just look at the sequence.
Remember, we made a huge
deal out of keeping it
straight the difference
between a sequence, the list,
and the series,
which is the sum.
I just look at the
sequence, 1 over k square,
and if that goes to 0 and is
decreasing, ding, ding, ding,
we're done.
Alternating series test
says the entire series,
the way it's written with
the alternator and all,
will converge.
And I get that by just
looking at the sequence.
Next step, I want to see
if it converges absolutely.
Well that means looking at
the series 1 over a k squared.
In other words, I
put up some values
around this whole
thing, which essentially
means ignoring the alternator.
In other words, treating
this as though all the terms
were positive.
So it means I'm looking at
not the sequence 1 over k
squared, but the series,
the sum, and the 1 over k
squared together.
So first step was
it's alternating,
so I just look at the sequence
1 over k squared, it went to 0,
we're done by the
alternating series test.
Now I look at the
series 1 over k squared
and I do whatever
I have to do to see
whether that converges or not.
All right?
We're back to sort of
hand-to-hand combat
with this series to
see what it does.
And any of the tests
we've used before
may be what we have to use.
This example, it's easy because
it's a p series, p equal to 2.
So it's going to converge.
That trips the switch that
says we converge absolutely.
So again, what's
the process when
you see an alternating series?
First I look at the sequence.
Is it decreasing and going to 0?
Yes, in this case.
If it wasn't then of
course it would divert
with the divergence test.
But if I just look at this
sequence, it's decreasing,
it goes to 0.
That's the first step.
If that happens then I
know my entire series
will at least converge.
Then I look at the
series 1 over k,
where my pointer is
for the bottom one.
The series, if it converges,
then I converge absolutely.
If it doesn't, I only
converge conditionally.
OK, let's look at something
that's not alternating.
I mentioned that the definition
of conditional convergence
and absolute divergence
didn't require
that we have an alternating
series, just that it makes
sense to talk about
whenever some of the terms
are positive or negative.
In fact, it makes sense to talk
about even if all the terms are
positive, it's just really
boring then because all
of the series that
we were doing back
in the other sections
with the comparison test,
with the ratio test, the root
test, where all the terms were
positive, all those were
converging absolutely.
We just never bothered to
say it because they all
were converging absolutely.
There's no conditional to
it because none of the terms
were ever negative.
In this example, wow,
I got cosine of k.
What in the world happens if I
start plugging 1, 2, 3, 4, 5,
6 into cosine?
Well, cosine will
always give us a number
between negative 1 and 1.
And when I start plugging
in integers, 1, 2, 3, 4, 5,
again notice this is in radians.
So 1 radian, 2 radians,
3 radians, and so on.
I'm going to get some
numbers that are positive,
some numbers that are
negative, and they're just
going to bounce around
all over the place.
So there isn't a nice plus,
minus, plus, minus, plus, minus
pattern.
There's just some positives,
some of them are negative.
Wow, OK.
This kind of example
shows up a lot.
See this very often
in the homework.
And the trick for
something like this,
especially when you
see a sine or a cosine,
it's usually sitting
there going, Hi I'm here
and I'm less than 1 and
bigger than negative 1.
I'm just sort of here to
randomly spew out minus signs.
So rather than try any of the
tests that we know before,
which don't work,
limit comparison tests,
ratio test, root test, integral
tests because some of the terms
are negatives.
So they don't work.
It's not an alternating series
test because some of the signs
are positive, some of
the signs are negative,
but they don't alternate
plus, minus, plus,
minus, they're just some are
positive and some are negative.
So what do we do?
Well, the trick is let's just
check from the very beginning
and see if it
converges absolutely.
Right?
So what we hope is, if we
just put absolute values
around this thing, that
it will still converge.
In which case it
converges absolutely.
And if it converges absolutely,
it absolutely converges.
It will definitely
converge and we're done.
And in this case
it's pretty easy
because the absolute value of
cosine is always less than 1.
So absolute value of
cosine of k over k squared
is always less than or
equal to 1 over k squared.
So what I'm really doing here
is the direct comparison test.
Right?
Because with the values around
it, all the terms are positive.
So the absolute
value of cosine k
over k squared, that's the
smaller sequence when I compare
that to 1 over k squared.
When I add up the
sequence 1 over k squared
and get a series, 1 over
k squared, it converges.
P series with p equal to 2.
So the original sequence
that I started with
converges absolutely
because it converges
when I put absolute values
around all the terms.
Now if this hadn't worked,
if the absolute value
trick around it
hadn't converged,
but it usually doesn't
happen because there's really
not a good way to play with
this unless something like this
happens.
One more bit about how
important it is for the terms
to be positive and
negative and so forth.
If you think back to things like
the ratio test, and the root
test, and the integral test,
and all those various tests,
they required that all the
turn would be positive.
Our textbook will at this
point list two more tests;
a generalized ratio
test and the same way
they do a generalized root test.
And all it means is this,
slap absolute values
around the sequence
that we're adding up,
slap up some values around the
ak, and then do the ratio test.
If it converges,
then the original
converged absolutely just
like in that last example.
You can play the same
game with the root test.
Slap absolutely values around
it, then do the root test.
If that converges with the
absolute values in there,
they will converge absolutely.
Why did they even
bother to say this?
Well because it's a textbook
and they always say everything.
But we'll need this idea when we
get to power series in a video
to come very shortly for you.
Whatever those power
series things are,
and you've heard me say that
word in some earlier videos,
we'll need this idea
of, hey, let's just
let slap absolute values
around it then do the test.
That will happen a
little bit more later.
and we'll need that idea.
So that's why they say it here.
This is one reason why when
you start listing out tests,
there are several of
them, but the book
acts like there's even
many more because they
stick generalized in
front of several of them
and give you even more tests.
