Alternating Double Series

Asked Apr 04 '20 at 03:58

Active Apr 04 '20 at 04:11

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Determine whether the series converges for $\frac{1}{2} < p \leq 1$ $$\sum_{k=2}^\infty \Bigl(\sum_{j=1}^{k-1} \frac{(-1)^k}{[j(k-j)]^p} \Bigr)$$

I've been stuck on this for a while now, and I'm not quite sure how to proceed with this. My assumption is that it does converge. However, the problematic thing is trying to prove the series satisfies the Alternating series test (the part where the term is eventually decreasing). How can I show that the term inside the bracket is decreasing? Or if this series diverges, can anybody give a hint? Thanks.

edited Apr 04 '20 at 04:11

user729424

5,061

asked Apr 04 '20 at 03:58

S10000

To use the alternating series test on the outer sum, you need to show the inner sum alternates in sign, so for odd $k$ it has one sign and for even $k$ the other. That seems reasonable, as there is a central peak in once case and not in the other, but I don't know how to do so. The inner sum is finite, so there is no convergence problem. – Ross Millikan Apr 04 '20 at 04:27
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Possible duplicate of Determine whether $S_n$ converges – Parcly Taxel Apr 04 '20 at 05:14

Alternating Double Series

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