Convergence/Divergence of series with terms $2^{n}\left ( \frac{n}{n+1} \right )^{n^{2}} $ and $\sin (n)\sin \frac{x}{n}$ Help me please with these 2 questions:
1.Does it converge or diverge? :
$$ \sum_{n=2}^{\infty }2^{n}\left ( \frac{n}{n+1} \right )^{n^{2}} $$
2.Check out absolute and conditional convergence of:  $x>0 $
$$ \sum_{n=1}^{\infty }\sin (n)\sin \frac{x}{n} $$
Thanks a lot!

Hint for 1:
For sufficiently large $n$, $(\frac{n}{n+1})^n = (1 - \frac{1}{n+1})^n \le c$ for some $ 0 \lt c \lt \frac{1}{2}$. Why?
Now trying using the above to prove that your series converges.
For part 2, I believe you can use the Dirichlet Test to prove convergence.
To show that the series does not converge absolutely, use $\sin (x/n) \ge x/2n$ for sufficiently large $n$ and use the fact that at least one of $n$, $n+1$ is more than $\frac{1}{2}$ away from the multiple of $\pi$ which is closest to them.
