comparison test - useful series to know what are some useful series to know for the comparison test along with their conditions?  I can think of the following:


*

*p-series

*geometric series

*harmonic series


are there want other series that are useful with the comparison test? 
thanks in advance
 A: Here is one that is useful to know, though not as commonly needed as those you list:
$$\text{If}\;\;p>1, \quad\sum_{n=2}^\infty\frac{1}{n(\ln n)^p}\;\;\text{converges}.
\text{ If}\;\;p\leq 1,\text{ the series diverges.}\tag{1}$$

$$\text{Also,}\;\;\sum_{n=0}^\infty \frac{1}{n!}\;\text{ converges. In fact,}\;\;
\sum_{n=0}^\infty\frac{1}{n!} = e.\tag{2}$$

Finally, the behavior of particular power series, and the corresponding radius of convergence of each, are good to know and understand.
A: In a general Calculus II course, your list is fine with just p-series and geometric series. Note that the harmonic series is just a p-series with $p=1$ which diverges. It is helpful, though perhaps trivial, to know that a constant of the harmonic series (say $\sum_{n=0}^{\infty}\frac{1}{2n} \equiv \frac{1}{2}\sum_{n=0}^{\infty}\frac{1}{n}$) diverges as well.
A helpful fact that when you're looking to use Comparison Test is to be careful with your inequalities and the way they go. Saying an expression is $\lt \infty$ is not very helpful for example in terms of determining divergence or convergence. In addition, for using the Limit Comparison Test, look at the behavior as $n\to\infty$ for your original series $a_n$ to determine a $b_n$ to use for the LCT.
You may also look for any series that corresponds to an improper integral whose convergence you know. The more series and improper integrals you know/figure out, the more you'll eventually have in your portfolio to use later on, which is helpful.
Useful facts (not necessarily for using Comparison Test though):
$$\lim_{n\to\infty} \frac{x^n}{n!} = 0 \tag{1}$$
$$\lim_{n\to\infty} \frac{n!}{x^n} = \infty \tag{2}$$
These two (though one is just an extension of the other) simply state that $n!$ grows faster than $x^n$ for $x \in \mathbb{R}$.
