$\displaystyle\sum \left(2^{-n} + \frac{3n+5}{n^3}\right)$ converges as a sum of three standard examples;
$\displaystyle\sum \left(2^{-n} + \frac{3n+5}{n^2}\right)$ diverges as a sum of three standard examples, two of which converge and one of which diverges;
$\displaystyle\sum \left(2^{-n} + \frac{3n+5}{n}\right)$ diverges by the Divergence Test;
$\displaystyle\sum \frac{1}{n^2+n}$ converges by comparison to $\displaystyle\sum \frac{1}{n^2}$;
$\displaystyle\sum \frac{n+\cos(n)}{n^3+n^2}$ also converges by comparison to $\displaystyle\sum \frac{1}{n^2}$, but we need the Limit Comparison Test;
$\displaystyle\sum \frac{1}{n \ln(n)}$ diverges by the Integral Test;
$\displaystyle\sum \frac{1}{n \left(\ln(n)\right)^2}$ converges by the Integral Test.
Examples (Part 2):
The video below shows how
$\displaystyle\sum \frac{(-1)^n}{\ln(n)}$ converges
(conditionally) by the Alternating Series Test;
$\displaystyle\sum \frac{(-1)^n(n+1)}{2n}$ is an alternating series, but it diverges, since the terms do not approach zero;
$\displaystyle\sum \frac{\cos(n)}{n^2}$ converges absolutely, by comparison to $\displaystyle\sum \frac{1}{n^2}$;
$\displaystyle\sum n^2\left(\frac23\right)^n$ converges by the Ratio Test;
$\displaystyle\sum \frac{(3/2)^n}{n^5}$ diverges by the Ratio Test;
$\displaystyle\sum \left ( 1 - \frac{1}{n^2} \right )^{n^3}$ converges by the Root Test.