$\sum_{n=1}^{\infty} \frac{(-1)^n \sqrt{n}}{n+1}$

$\sum_{n=1}^{\infty} \frac{n^2}{(n+1)!}$

$\sum_{n=1}^{\infty} \frac{1}{n(\ln n)^{1/2}}$ for $n \geq 2$

$\sum_{n=1}^{\infty} \frac{n^2 + 1}{n^3 + 2n}$