# How do you determine the convergence or divergence of sum_(n=1)^(oo) cosnpi?

Feb 7, 2017

The series:

${\sum}_{n = 1}^{\infty} \cos n \pi$

is indeterminate

#### Explanation:

A necessary condition for a series to be convergent is that the sequence of its terms is infinitesimal, that is ${\lim}_{n \to \infty} {a}_{n} = 0$

We have:

$\cos n \pi = {\left(- 1\right)}^{n}$

so the series is not convergent.

If we look at the partial sums we have:

${s}_{1} = - 1$

${s}_{2} = - 1 + 1 = 0$

${s}_{3} = 0 - 1 = - 1$

$\ldots$

Clearly the partial sums oscillate, with all the sums of even order equal to zero and all the sums of odd order equal to $- 1$.

Thus, there is no limit for $\left\{{s}_{n}\right\}$ and the series is indeterminate.