# Does the series $\sum \limits _{n=0}^{\infty} \cos(n\pi)$ converge or diverge?

Does the series $\sum \limits _{n=0}^{\infty} \cos(n\pi)$ converge or diverge?

On substituting values I get alternate $1$ and $-1$.

So taking sum of infinite GP, I get $\dfrac 1 2$. So it looks like it converges.

• Try to write out some values of $a_n$. What do you see? Nov 26, 2015 at 18:52
• @NigelOvermars I get altternate 1 and -1. Nov 26, 2015 at 18:53
• @user293702 So what does that tell you? Nov 26, 2015 at 18:56
• @NigelOvermars I edited my question Nov 26, 2015 at 18:58
• @user293702 Are you talking about the sequence $\{a_n\}$ or the sum $S_n = a_1 + ... + a_n$? Nov 26, 2015 at 19:01

The sequence $\{a_n\}$ defined by $a_n = \cos(n \pi)$ diverges, since we have $a_{2n} = 1$ and $a_{2n+1} = -1$ for every $n \in \mathbb{N} \cup \{ 0 \}$. Hence two different subsequences have different limits, which implies divergence .
The series (sum) $S_n = a_1 + ... + a_n$ also diverges, since we have, following the same line of reasoning, $S_{2n} = 0$ and $S_{2n+1}=-1$. Hence the series also diverges.