Why does [math]\displaystyle\sum_{n = 1}^{\infty}2^{-\frac{n^2}{n+1}}[/math] converge but [math]\displaystyle\sum_{n = 1}^{\infty}2^{-\frac{n}{n^2+1}}[/math] diverge?
Is [math]\displaystyle\sum_{n=1}^{\infty} 2 \sin\left(\frac{2}{n}\right) [/math] convergent or divergent?