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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

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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.

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+1. The series $\displaystyle \sum_{k=2}^{\infty} \dfrac1{k\log^p(k)}$ is often useful and can help us to analyze/ get an idea for slightly non-trivial convergence/divergence. – user17762 Nov 29 '12 at 1:15
More generally $$\sum_{k=N}^\infty \frac{1}{k^{a_0}(\log k)^{a_1}(\log^{\circ2}k)^{a_2} \dotsb (\log^{\circ n}k)^{a_n}},\quad N \text{ sufficiently large}$$ converges when $a_0=a_1=\dotsb=a_{n-1}=1$ and $a_n >1$. – kahen Nov 29 '12 at 14:30

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 $x \in \mathbb{R}$.

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