Let $a_n=\frac{(-1)^n}{\sqrt{(1+n)}}$ then which of the following is correct?

Question

$a_n=\frac{(-1)^n}{\sqrt{(1+n)}}$ the $a_n$ is convergent by leibniz rule for alternating series. How can I test the convergence or divergence of $c_n$

Answer 1

$c_n$ diverges since $|c_n|=|\sum_{k=1}^{n}a_ka_{n-k}|=|\sum_{k=1}^{n}(-1)^n\frac{1}{\sqrt{(1+k)(1+n-k)}}|=\sum_{k=1}^{n}\frac{1}{\sqrt{(1+k)(1+n-k)}}>\sum_{k=1}^n\frac{1}{n}=1$.