If $a_{n} = ((-1)^{n})/(\sqrt{1+n})$ then is $\sum_{k=0}^{n} (a_{n-k})(a_{k})$ convergent? If $a_{n} = ((-1)^{n})/(\sqrt{1+n})$ then is $C_{n} = \sum_{k=0}^{n} (a_{n-k})(a_{k})$ convergent?
$a_{n}$ is conditionally convergent, but how do I check the convergence of $C_{n}$?
I could transform $C_{n}$ in following way 
$C_{n} = \sum_{k=0}^{n} \frac{(-1)^{n}}{(1+k)^{1/2}(1+n-k)^{1/2}}$ 
How to proceed?

Hint:
$$\frac{1}{\sqrt{k+1}\sqrt{n-k+1}} > \frac{1}{\sqrt{n+1}\sqrt{n-0+1}} = \frac{1}{n+1} \\ \implies \sum_{k=0}^n \frac{1}{\sqrt{k+1}\sqrt{n-k+1}} >  \ldots$$
and $|C_n| \not\to 0 \implies \lim_{n \to \infty} (-1)^n |C_n|$ DNE
