Determine the convergence of the series (-1)ⁿ n² / (1+n²)

by RoRi

March 10, 2016

Consider the series

$\sum_{n=1}^{\infty} (-1)^n \frac{n^2}{1+n^2}.$

Determine whether the series converges or diverges. If it converges, determine whether it converges conditionally or absolutely.

The given series diverges.

Proof. The series diverges since the limit

$\lim_{n \to \infty} (-1)^n \frac{n^2}{1+n^2} \neq 0.$

(Since the limit of the even terms is 1 and the limit of the odd terms is $-1.) \qquad \blacksquare$

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