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

Jan 3, 2017

${\sum}_{n = 1}^{\infty} {\left(- 1\right)}^{n} {n}^{2} / \left({n}^{2} + 1\right)$ does not converge

#### Explanation:

This is an alternating series, so the necessary condition for it to converge is that:

$\lim {a}_{n} = 0$

${a}_{n + 1} / {a}_{n} < 1$

As:

${a}_{n} = {n}^{2} / \left({n}^{2} + 1\right)$

we have:

${\lim}_{n} {a}_{n} = {\lim}_{n} {n}^{2} / \left({n}^{2} + 1\right) = 1$

Therefore the series does not converge.