If $\lim\limits_{n \to \infty} n^2(a_n)^2=5$, does $\sum_{n=1}^{\infty} a_n$ converge or diverge?

Question

If the limit given is equal to 5, then the series $\sum_{n=1}^{\infty} n^2(a_n)^2$ diverges. Also, the series $\sum_{n=1}^{\infty} (a_n)^2$ converges through limit comparison test. But I don't know how to tie that in with $\sum_{n=1}^{\infty} a_n$.

Answer 1

It may converge Or diverge. To see this consider the following two cases:

$${{a}_{n}}=\frac{\sqrt{5}}{n}$$ $${{a}_{n}}={{\left( -1 \right)}^{n}}\frac{\sqrt{5}}{n}$$