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The comparison test, again.

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[MUSIC]

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Well, here's something that happens.

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Well, maybe you're analyzing a couple
series, and

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the first thing that you do is try to

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apply the limit test, and you find that
both

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cases, the limit of the nth term is zero.

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So, at least these series aren't diverging
for an obvious

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reason, like the limit of the nth term is
non zero.

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So you got two series that may or may not
converge.

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But, even though the limit

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of a sub n is zero, and the limit of b sub
n is zero, it could be that the limit as n

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goes to infinity of a sub n over b sub n
might be some number L, which is positive.

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What does that really mean?

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Well one way to think about that is that
it's sort of saying something like this.

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It's saying that a sub n, b sub n are
almost multiples of

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each other, like a sub n is almost a
multiple of b sub n,

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at least when n is really big.
I can be more precise with epsilons.

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So let's set epsilon equal to L in this
case.

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Epsilon's going to be a positive number,
but

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I'm assuming that my limit, L, is
positive.

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So let's set epsilon equal to L.
Then the definition of limit says what?

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It says that there's some big N, so that
whenever little

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n is greater than or equal to big n the
distance

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between the thing I'm taking the limit of,
a sub n

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over b sub n, and my limit, L, is less
than epsilon.

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And in this case, right, epsilon is L.

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That lets me compare a sub n and b sub n.
Well, how so?

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Let me make this assumption that the a sub
n's and the b sub n's are non-negative.

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I'm going to want that because I'm going
to apply the comparison test in a moment.

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So I can simplify this a bit.

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Instead of making this claim, I can just
get

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rid of the absolute value bars, it's still
true, right.

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A sub n over b sub n, minus L is less than
L.

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You can add L to both sides of that
inequality and I get

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this, that a sub n over b sub n, is less
than 2L.

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And now I can multiply both sides by b sub
n, and that's okay,

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because b sub n is non-negative, so

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it doesn't change the direction of this
inequality.

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And that

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tells me that at least for large values of
little n, a sub n is less than 2L b sub n.

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Now all these pieces are really setting up
a comparison test.

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How is that going to work?

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I got to remember this is only true for
large N, but that's okay.

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Well suppose that I knew that this series,
the sum little

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n goes from 1 to infinity of b sub n
converged,

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well then I would know that this series,
the sum little n

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goes from 1 to infinity of 2L b sub n also
converges.

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Right, I can multiply a convergent series
just by some number.

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But now I'm in the position to apply the
comparison test.

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Granted this statement's only true for
large values of little n but that's

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okay, right, because convergence only
depends on a tail, so this statement

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then implies that this series, the sum of
the a sub n's converges, because

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this is bigger than a sub n, I mean at
least for large values of little n.

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So I'm getting a theorem that's telling me
if I've got two

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series of non-negative terms and this
series

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converges, then this series converges,
provided that

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this is true.
Let me summarize that.

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So, if I've got a sub n's are all
non-negative, b sub n's are all

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non-negative, the sum of the b sub n's
converges, and this limit

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statement that the limit of the ratios
between the a sub n's and

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the b sub n is some finite value L, which
is positive, then I

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can conclude that this series, the sum of
the a sub n's, n goes from 1 to

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infinity, converges as well.
Now let me exchange the roles of a and b.

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So I'd like to be able to start with

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the assumption that the series of the a
sub n's

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converges and then conclude that the
series of the b

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sub n's converges, but actually that's the
exact same statement.

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All right, watch this.
If I just replace this limit with this

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limit, now I'm in the exact same
situation, except now b

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sub n's and a sub n's roles are exchanged,
all right, and that means that if I know

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that this series converges, then I know
that this series converges as well.

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Now put it all together.

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Well, since the convergence of a sub n
implies the

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convergence of b sub n, and the

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convergence of b sub n implies the
convergence

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of a sub n in the presence

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of this limit statement, equivalently,
this limit statement.

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I can simply this a bit, I can say that

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if I've got two sequences of numbers, a
sub n and

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b sub n, both non negative, and this limit
exists

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and is equal to some number bigger than
zero, then this

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series converges, if and only if, this
series converges.

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The sum of the b sub n's converges if and
only if, the sum a sub of n's converges.

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This convergence test has a name.

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This is called The Limit Comparison Test,
and it's

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one of the situations when two series, in
this case

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the sum of the b sub n's and the sum of
the a sub n's, share the same fate.

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They either both converge or they both
diverge.

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