Answer
See below.
Work Step by Step
These are given by Theorems 10 (Direct comparison) and 11 (Limit comparison) in section 4.
The direct comparison is applicable for series with nonnegative terms.
Say that we have two such series, and $a_{n}\leq b_{n}$, for all $n.$
If $\displaystyle \sum_{n=1}^{\infty}a_{n} $ diverges, then, since $\displaystyle \sum_{n=1}^{\infty}b_{n}$ has greater terms, it will diverge too.
If $\displaystyle \sum_{n=1}^{\infty}b_{n} $ converges, since $a_{n}\leq b_{n}$, the series $\displaystyle \sum_{n=1}^{\infty}a_{n}$ also converges.
(If the "lesser" diverges, the "greater" diverges too, and if the "greater" converges, the "lesser" will also converge.)
The limit test is applicable for a series with positive (nonzero) terms.
Comparing limits of ratios of terms of two such series, we have:
If $\displaystyle \lim_{n\rightarrow\infty}\frac{a_{n}}{b_{n}}=$nonzero constant, $\displaystyle \sum_{n=1}^{\infty}a_{n}$ and $\displaystyle \sum_{n=1}^{\infty}b_{n}$ both either converge or diverge.
If the limit is $0$, then the convergence of $\displaystyle \sum_{n=1}^{\infty}b_{n}$ implies the convergence of $\displaystyle \sum_{n=1}^{\infty}a_{n}$.
If the limit is $\infty, $ then the divergence of $\displaystyle \sum_{n=1}^{\infty}b_{n}$ implies the divergence of $\displaystyle \sum_{n=1}^{\infty}a_{n}$.
If we can't easily apply other tests, then we "go hunting" for a series to which we can compare our series, in order to determine its convergence.
