I'm having trouble finding r in these two problems.
$\displaystyle \sum_{n=1}^{\infty} {\frac{n}{n+1}}$
and
$\displaystyle \sum_{n=0}^{\infty} {sin^n(\frac{\pi}{4}+n{\pi})}$
So is the question meant to be "Find whether the following series converge or diverge"? You've already been told the answer to the first - the reason is that $\displaystyle \lim_{n \to +\infty} \frac{n}{n + 1} \neq 0$.
It would have saved the time of everyone if you did not provide a misleading post title.