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The tail is what matters.

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[MUSIC]

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If you go out and read the literature

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on convergence tests and the like, you'll
sometimes find

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that those theorems are written in a way

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that's a bit more vague than we're used
to.

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The Comparison Test.

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If a sub n is between 0 and b sub n for
all n, and the series sum of the

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b sub n's converges, then this series, the
sum of the a sub n's converges as well.

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Do you see what's missing?
Well, I didn't write the sum n goes from 1

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to infinity of b sub n, or the sum n goes
from

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1 to infinity of a sub n.
But this is actually okay.

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here's a theorem.

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Let M be a natural number then this
series, the

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sum m goes from M to infinity of a sub m,

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converges, if and only if this series, the
sum little m goes from 1 to

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infinity of a sub m, converges.
In short, convergence depends on the tail.

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So I've got this series, the sum little m
goes from 1 to infinity of a sub m.

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And I could start writing it out like
this.

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It's a sub 1 plus a sub 2, blah, blah,
blah,

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plus a sub M minus 1 plus a sub M plus a

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sub M plus 1 plus and so on.
And I can take out my scissors of

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mathematics and cut this series like this.

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This piece here I'll call the head, and
what's left over, which is really the

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sum m goes from M to infinity of a sub m,
this is the tail.

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And what the theorem tells me is that this
series converges if and

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only if this series converges So converges
only depends on the tail.

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Why does convergence only depend on the
tail?

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Well, let's suppose that this tail
converges, that

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means something of the sequence of partial
sums.

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What that means is, that if I consider the

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sequence of partial sums, s sub n equals
the sum

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m from M to n of a sub m.
That sequence of partial

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sums converges to something, I'll call it
L, it's the value of this series.

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I can relate those partial sums to the
partial

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sums for the series that begins with a sub
1.

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So now let's think about this series.

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Right, this series m goes from one to
infinity of a sub m.

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The value

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of this series is by definition the limit
of the partial sums for that series, and

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that's the limit n goes to infinity of the
sum little m goes from 1 to n of a sub m.

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Now I can analyze this in terms of the
series that starts at M.

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Alright?

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What is this limit?

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Well, that limit is a sub 1 plus dot, dot,
dot, plus a sub M minus

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1 plus this limit.

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The limit n goes to infinity of the sum n
from M to n of a sub M.

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Alright, both of these are just the limits
of the

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sum of the terms, a sub 1 through a sub n.

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But now, I'm assuming that this limit
exists because I'm assuming

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that that, that series that starts at M
converges as the limit

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of the partial sums for that series which
I was calling L before, which is just to

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say that this series has a finite value,
right, this limit exists.

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And what that means, is that if I suppose
that the tail

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of a series converges, then the original
series converges as well.

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Here's the upshot.
If you only care about

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convergence, writing down this, just the
sum over n

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of a sub n, is just as good as writing

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down this more complicated looking thing,
the sum little

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n goes from 1 to infinity of a sub n.

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Because if all I care about is
convergence,

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I don't actually need to know where the
series

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is starting, because regardless of where I
start

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the series, they all converge or they all
diverge.

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It only depends on the tail.

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