# How do you test the alternating series Sigma (-1)^n from n is [1,oo) for convergence?

##### 1 Answer
Jan 30, 2017

the series is indeterminate.

#### Explanation:

We can easily see that the series is not convergent, since:

${\lim}_{n \to \infty} {\left(- 1\right)}^{n} \ne 0$

We can take a closer look at the partial sums:

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

${s}_{1} = - 1$
${s}_{2} = 0$
${s}_{3} = - 1$
$\ldots$

and we can prove by induction that:

$\left\{\begin{matrix}{s}_{2 n} = 0 \\ {s}_{2 n + 1} = - 1\end{matrix}\right.$

so that partial sums oscillate between the two values and do not converge to a limit.