How To Determine If $\sum_{n=1}^{\infty}\frac{(-1)^n}{n}\left(\sum_{k=0}^{n-1}\binom{2k}{k}\binom{k}{n-k}\right)$ Converges or Diverges?

Question

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

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.

Answer 1

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?