Let be a convergent series in which . Prove that the series converges as well.

How should I start?

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- Oct 13th 2013, 07:56 PMvidomagruProving that (1/n)(a_n + a{n+1} + ... + a_{1n-1}) converges.
Let be a convergent series in which . Prove that the series converges as well.

How should I start? - Oct 13th 2013, 08:29 PMvotanRe: Proving that (1/n)(a_n + a{n+1} + ... + a_{1n-1}) converges.
- Oct 13th 2013, 08:50 PMSlipEternalRe: Proving that (1/n)(a_n + a{n+1} + ... + a_{1n-1}) converges.
- Oct 14th 2013, 06:58 PMvidomagruRe: Proving that (1/n)(a_n + a{n+1} + ... + a_{1n-1}) converges.
- Oct 14th 2013, 07:39 PMSlipEternalRe: Proving that (1/n)(a_n + a{n+1} + ... + a_{1n-1}) converges.
Let . Since the series converges absolutely, the sequence converges (and is Cauchy)[/tex]. Rewriting the series you are given:

Writing out a few elements of this series, you have

Collecting like terms, we have

So, the general form of the series seems to be . Use induction to prove that this is true, and then use the Alternating Series Test to prove that it converges conditionally. - Oct 15th 2013, 07:21 PMvidomagruRe: Proving that (1/n)(a_n + a{n+1} + ... + a_{1n-1}) converges.
- Oct 15th 2013, 07:23 PMSlipEternalRe: Proving that (1/n)(a_n + a{n+1} + ... + a_{1n-1}) converges.
Hmm, that's a good point. Maybe you can show that for any , there exists such that for all , the partial sum of the first terms are no more than apart?

- Oct 15th 2013, 07:34 PMvidomagruRe: Proving that (1/n)(a_n + a{n+1} + ... + a_{1n-1}) converges.
I am not quite sure how to apply that, but I thought of something else instead that might be helpful. Isn't a subsequence of which is convergent so any subsequence would also be convergent?

Or because is a partial sum of a convergent series , also converges.

Then we would just need to apply . Not sure how to do that? - Oct 15th 2013, 08:10 PMSlipEternalRe: Proving that (1/n)(a_n + a{n+1} + ... + a_{1n-1}) converges.
A subsequence is very different from a partial sum. A sum of partial sums may not converge. But, I think I figured it out.

Now, can you show that for almost all ?

Since converges and for all n, it must be that . So what I am getting at, - Oct 15th 2013, 08:22 PMvidomagruRe: Proving that (1/n)(a_n + a{n+1} + ... + a_{1n-1}) converges.
- Oct 15th 2013, 08:22 PMSlipEternalRe: Proving that (1/n)(a_n + a{n+1} + ... + a_{1n-1}) converges.
Also, writing out the terms of the series, we have:

Can you show that the coefficient of is always no greater than 1? - Oct 15th 2013, 08:29 PMvidomagruRe: Proving that (1/n)(a_n + a{n+1} + ... + a_{1n-1}) converges.
- Oct 15th 2013, 08:31 PMvidomagruRe: Proving that (1/n)(a_n + a{n+1} + ... + a_{1n-1}) converges.
- Oct 15th 2013, 08:36 PMSlipEternalRe: Proving that (1/n)(a_n + a{n+1} + ... + a_{1n-1}) converges.

shows up in the first term only. shows up in the second term only. shows up in terms 2 and 3. shows up in terms 3 and 4. shows up in terms 3, 4, and 5. shows up in terms 4, 5, and 6. Seeing a pattern?

Also, this expansion should show you how each term is a sum of n elements over n. - Oct 15th 2013, 08:55 PMvidomagruRe: Proving that (1/n)(a_n + a{n+1} + ... + a_{1n-1}) converges.