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$$\sum_{n=1}^{\infty}\frac{(-1)^n}{n}\left(\sum_{k=0}^{n-1}\binom{2k}{k}\binom{k}{n-k}\right)$$

Question : How do i determine if the above Series Converges to Diverges?

I have no idea where to begin since i do not have much experience with Sums of this type.

Thank you kindly for your help and time.

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    $\begingroup$ Are you sure that the upper bound of the inner sum is $n$ ? Could it be $(n-1)$ ? $\endgroup$ – Claude Leibovici Jun 1 at 3:20
  • $\begingroup$ @ClaudeLeibovici yes it is $(n-1)$ my apologies, how did you know? $\endgroup$ – No-one Important Jun 1 at 3:31
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    $\begingroup$ Do you want to see $\binom{n}{0}$ ? $\endgroup$ – Claude Leibovici Jun 1 at 3:36
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Here's a hint:

The inside series is a sum of binomial coefficients, so it ``looks like it should be big.'' To this end, we can try to show that it's greater than, say, $n$; if this is true, then the terms of the series cannot tend to $0$.

By looking at some terms of the inside series, can you see why this should be the case?

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