# Convergence or divergence of when the alternating series test fails.

I want to investigate the convergence or divergence of the following series:$$\sum _{n=0}^{\infty }\:\frac{\left(n!\right)^2}{\left(2n\right)!}(-4)^n$$ However, the alternating series test fails because the absolute value of the ratio of successive is $$\frac{4n+4}{4n+2}$$ which means the sequence $$\{|a_n|\}$$ is increasing and so the alternating series test does not apply. I have proven that this series does not converge absolutely, because $$n\frac{a_n}{a_{n+1}}-n=\frac{-2n}{4n+4}\leq 0$$ so $$\sum^\infty|a_n|$$ diverges by Raabe's test.

But how do I know if the series might still converge conditionally?

I also tried the root test, and it turns out that $$\limsup_{n\rightarrow\infty}\sqrt[n]{|a_n|}=\lim_{n\rightarrow\infty}\sqrt[n]{\frac{4^n\left(n!\right)^2}{\left(2n!\right)}}=1$$ so that test was also inconclusive. What else can I try?

If $$\lvert a_n\rvert$$ is increasing and nonzero, then $$a_n\not \to 0$$, and therefore $$\sum_{n=0}^\infty a_n$$ does not converge.