$c_n = \sum_{k=0}^n a_k.a_{n-k}$ where $a_n=\frac{(-1)^n}{\sqrt{1+n}}$.
Check whether the sum $\sum_{k=0}^nc_n$ is convergent or not.
My attempt:
$c_n = a_0.a_n + a_1.a_{n-1} + \cdots + a_n.a_0$
Now $c_n = a_0.(s_n-s_{n-1}) + a_1(s_{n-1}-s_{n-2}) + \cdots + a_n.s_0$
$c_{n-1} = a_0.(s_{n-1}-s_{n-2}) + a_1(s_{n-2}-s_{n-3}) + \cdots + a_{n-1}.s_0$
I think $\sum_{k=0}^n c_n = a_0.(s_n) + \cdots + a_n.s_0$
I cant proceed after this..