Which of the following series will converge and which one will diverge? Can anyone help me out that which of the following series will converge and which one will diverge, with some explanation?
A) $\sum_{n=1}^\infty \sin\left(\frac{\pi}n\right)$ 
B) $\sum_{n=1}^\infty (-1)^n  \cos\left(\frac{\pi}n\right)$
Thanks a lot.

Euler said that if $x$ is an infinitely small positive number than $\sin x=x$.  Today we way $\lim\limits_{x\to0}\dfrac{\sin x}x=1$.  At any rate we know that if $x$ is a sufficiently small positive number then
$$
\frac x 2 <\sin x < x,
$$
so
$$
\frac\pi2\sum_n \frac 1 n =\sum_n \frac\pi{2n}\le\sum_n \sin\frac \pi n \le \sum_n \frac \pi n = \pi\sum_n \frac 1 n.
$$
Can you do the rest?
As for the other series, the terms don't approach $0$.
