# How do you determine whether the infinite sequence a_n=n*cos(n*pi) converges or diverges?

Sep 24, 2014

The sequence ${a}_{n} = n \cos \left(n \pi\right)$ diverges.

Let us look at some details.

Let us take the limit of its subsequence of only even terms.

$\setminus {\lim}_{n \to \infty} {a}_{2 n} = {\lim}_{n \to \infty} \left(2 n\right) \cos \left(2 n \pi\right)$

since $\cos \left(2 n \pi\right) = 1$ for all integer $n$,

$= {\lim}_{n \to \infty} \left(2 n\right) = 2 \left(\infty\right) = \infty$

Since the subsequence diverges, the original sequence also diverges.