let all xn>=0 for all natural numbers n. Assume that sigma(n from 1 to infinity) xn<infinity

Prove that the lim as n tends to infinity of xn=0

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- Nov 12th 2009, 09:25 PMamm345Limit Proof
let all xn>=0 for all natural numbers n. Assume that sigma(n from 1 to infinity) xn<infinity

Prove that the lim as n tends to infinity of xn=0 - Nov 12th 2009, 10:07 PMredsoxfan325
- Nov 12th 2009, 10:34 PMamm345
I'm sorry, I'm still having troubles seeing where to go next?

Am I looking to end with a contradiction? - Nov 12th 2009, 10:36 PMredsoxfan325
Yes. Since is a constant, you can pull it out of the sum.

which clearly diverges to . This is a contradiction, so . - Nov 13th 2009, 02:18 AMhjortur
I just wanted to show another way to do this.

Let

You know that

So taking the limit:

But because you know that the sum is finite:

- Nov 13th 2009, 06:03 AMHallsofIvy
Yet another way: Let be the nth partial sum. Then the series converges if and only if the sequence of partial sums converges (that's the

**definition**of convergence of a sequence).

But then, if does not go to 0, does not go to 0. That means that cannot go to 0 as n and m go to 0 independently and so is not a Cauchy sequence.

Since every convergent sequence is a Cauchy sequence, does not converge and thus does not converge.