# How do you determine the convergence or divergence of Sigma ((-1)^(n+1)n^2)/(n^2+5) from [1,oo)?

The non-alternating portion of the sequence that makes up the series is ${n}^{2} / \left({n}^{2} + 5\right)$.
Note that ${\lim}_{n \rightarrow \infty} {n}^{2} / \left({n}^{2} + 5\right) = 1$.
This means that as $n$ grows infinitely larger, the series will begin to resemble ${\sum}_{n = 1}^{\infty} {\left(- 1\right)}^{n}$, which diverges because it never "settles".
A powerful tool for testing divergence is the rule that if $\sum {a}_{n}$ converges, then ${\lim}_{n \rightarrow \infty} {a}_{n} = 0$. That is, as $n$ gets larger, the terms of any convergent series will approach $0$.
Here, ${\lim}_{n \rightarrow \infty} {a}_{n} = 1$, so the series is divergent.