For $1$ through $4,$ determine whether the series is convergent or divergent.Show all your work in full detail for full credit.

${\displaystyle \underset{n=1}{\overset{}{}}}n{e}^{-n^2}$

${\displaystyle \underset{n=1}{\overset{}{}}}\frac{1}{n^{22}+4}$

${\displaystyle \underset{n=1}{\overset{}{}}}\frac{2^n}{1+5^n}$

${\displaystyle \underset{n=1}{\overset{}{}}}\frac{n^4+n^2+7}{n^6+n^4}$