That is a matter of observation.
Generally,
Try the ratio test when dealing with factorials or kth powers.
The Root Test when dealing with kth powers as well.
The Limit Comparison Test is easier to apply then the Comparison Test, but still requires some skill in choosing the series \(\displaystyle \sum b_{k}\) for comparison.
Let \(\displaystyle \sum a_{k}\) and \(\displaystyle \sum b_{k}\) be series with positive terms such that we have \(\displaystyle {\rho}=\lim_{k\to \infty}\frac{a_{k}}{b_{k}}\)
If \(\displaystyle 0<{\rho}<\infty\), then both series converge or both diverge.
The Alternating Series Test for, of course, alternating series.
