The tail is what matters.
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If you go out and read the literature
on convergence tests and the like, you'll sometimes find
that those theorems are written in a way
that's a bit more vague than we're used to.
The Comparison Test.
If a sub n is between 0 and b sub n for all n, and the series sum of the
b sub n's converges, then this series, the sum of the a sub n's converges as well.
Do you see what's missing? Well, I didn't write the sum n goes from 1
to infinity of b sub n, or the sum n goes from
1 to infinity of a sub n. But this is actually okay.
here's a theorem.
Let M be a natural number then this series, the
sum m goes from M to infinity of a sub m,
converges, if and only if this series, the sum little m goes from 1 to
infinity of a sub m, converges. In short, convergence depends on the tail.
So I've got this series, the sum little m goes from 1 to infinity of a sub m.
And I could start writing it out like this.
It's a sub 1 plus a sub 2, blah, blah, blah,
plus a sub M minus 1 plus a sub M plus a
sub M plus 1 plus and so on. And I can take out my scissors of
mathematics and cut this series like this.
This piece here I'll call the head, and what's left over, which is really the
sum m goes from M to infinity of a sub m, this is the tail.
And what the theorem tells me is that this series converges if and
only if this series converges So converges only depends on the tail.
Why does convergence only depend on the tail?
Well, let's suppose that this tail converges, that
means something of the sequence of partial sums.
What that means is, that if I consider the
sequence of partial sums, s sub n equals the sum
m from M to n of a sub m. That sequence of partial
sums converges to something, I'll call it L, it's the value of this series.
I can relate those partial sums to the partial
sums for the series that begins with a sub 1.
So now let's think about this series.
Right, this series m goes from one to infinity of a sub m.
The value
of this series is by definition the limit of the partial sums for that series, and
that's the limit n goes to infinity of the sum little m goes from 1 to n of a sub m.
Now I can analyze this in terms of the series that starts at M.
Alright?
What is this limit?
Well, that limit is a sub 1 plus dot, dot, dot, plus a sub M minus
1 plus this limit.
The limit n goes to infinity of the sum n from M to n of a sub M.
Alright, both of these are just the limits of the
sum of the terms, a sub 1 through a sub n.
But now, I'm assuming that this limit exists because I'm assuming
that that, that series that starts at M converges as the limit
of the partial sums for that series which I was calling L before, which is just to
say that this series has a finite value, right, this limit exists.
And what that means, is that if I suppose that the tail
of a series converges, then the original series converges as well.
Here's the upshot. If you only care about
convergence, writing down this, just the sum over n
of a sub n, is just as good as writing
down this more complicated looking thing, the sum little
n goes from 1 to infinity of a sub n.
Because if all I care about is convergence,
I don't actually need to know where the series
is starting, because regardless of where I start
the series, they all converge or they all diverge.
It only depends on the tail.
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