# How do you test for convergence given Sigma (-1)^n(1-1/n^2) from n=[1,oo)?

Jan 27, 2017

The series:

${\sum}_{n = 1}^{\infty} {\left(- 1\right)}^{n} \left(1 - \frac{1}{n} ^ 2\right)$

is not convergent.

#### Explanation:

A necessary condition for any series to converge is that the general term of the succession is infinitesimal, that is:

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

In our case we have:

${\lim}_{n \to \infty} \left(1 - \frac{1}{n} ^ 2\right) = 1$

and so:

${\lim}_{n \to \infty} {a}_{n} = {\lim}_{n \to \infty} {\left(- 1\right)}^{n} \left(1 - \frac{1}{n} ^ 2\right)$

is indeterminate.