Does the series $\frac{\cos(\pi n)}{n}$ converge?

Question

Does the series $\left\{{\cos(\pi\cdot n)\over n}\right\}$ converge?

I think it does, but can't find a series convergence test that applies to it.

I can't compare it to the series $\left\{{1\over n}\right\}$, and I can't use the integral test. What do I do?

Answer 1

The criteria you mention apply only for series with positive terms, which is not the case here. $$\sum_n\frac{\cos n\pi}n=\sum_n\frac{(-1)^n}n$$ This is the alternating harmonic series, it it is proved to be convergent using Leibniz criterion for alternating series:

If $f(n)$ decreases to $0$, the alternating series $\displaystyle\sum_n (-1)^nf(n)$ converges.

Answer 2

Hint : $\cos(\pi n) = (-1)^n$ and $\frac{1}{n}-\frac{1}{n+1} = \frac{1}{n(n+1)}$