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

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Answer 1 · 2017-01-03T15:16:50

$sum_(n=1)^oo (-1)^n n^2/(n^2+1)$ does not converge

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:

$a_n = n^2/(n^2+1)$

we have:

$lim_n a_n = lim_n n^2/(n^2+1) = 1$

Therefore the series does not converge.

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