Examine each of the following series and determine whether or not they converge.

Give reasons in each case.

$(i)$

$(i i)$

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

(iii)

$$\sum_{n=1}^{\infty} \frac{n^{3}+(-1)^{n} 8 n^{2}+1}{n^{4}+(-1)^{n+1} n^{2}}$$

$(i v)$

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