$\sum \frac{1}{n^2}$ with $\lim = 1$ → converges

$\sum \frac{1}{n^2}$ with $\lim = 0$ → inconclusive

$\sum \frac{1}{n}$ with $\lim = 0$ → inconclusive

$\sum \frac{1}{n^3}$ with $\lim = 1$ → converges