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
what are some useful series to know for the comparison test along with their conditions? I can think of the following:
are there want other series that are useful with the comparison test?
thanks in advance
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.
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 $x \in \mathbb{R}$.