Determine whether the following series converge or diverge. Any tests that you use should be carefully stated.

(a)

$$\sum_{n \geqslant 1} \frac{n !}{n^{n}}$$

(b)

$$\sum_{n \geqslant 1} \frac{1}{n+(\log n)^{2}}$$

(c)

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

(d)

$$\sum_{n \geqslant 1} \frac{(-1)^{n}}{n\left(2+(-1)^{n}\right)}$$