# Useful series for limit comparison test

I already know the harmonic series ($1/n$, which diverges) and following the link, $1/n^2$ that converges.

What other useful series could you teach me, or perhaps some general advice? Any website/resource is very welcome also, since most of my search for them reveals only the harmonic one and not much more. (I know how to execute the test!).

Have a nice day!

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Standard series for which convergence is known are: geometric series; $p$-series; any series that corresponds to an improper integral whose convergence you know; powers over factorials (e.g., $\frac{2^n}{n!}$), etc. The more series you figure out, the more you'll have in your portfolio to use for comparisons. – Arturo Magidin Feb 17 '11 at 19:28
Thanks for your share Arturo! – Qosmo Feb 17 '11 at 19:47

There is the generalization of what you put above, the so called $p$-series. Let $p\in \mathbb{R}$, then $$\sum_{n=1}^\infty \frac{1}{n^p}$$ converges if and only if $p>1$. This can be proven using the integral test mentioned by Shai Covo.

Also of interest are alternating series, such as $$\sum_{n=1}^\infty \frac{(-1)^n}{\sqrt{n}}$$ since there are good tests to see if they converge. (The above sum converges) Specifically, the alternating series test tells us that if we have a sequence $a_n$ with

(1) $a_n\cdot a_{n+1} <0$ for every $n$ (it alternates signs)

(2) $|a_{n+1}|\leq |a_n|$

(3) $\lim_{n\rightarrow \infty} a_n =0$

Then $\sum_{n=1}^\infty a_n$ converges. This then brings up the topic of Conditional and Absolute convergence. For a generalization of the alternating series test, see Dirichlets Test. (This test allows us to give the conditions of convergence for series such as $\sum_{n=1}^\infty a_n \sin (n)$)

Hope that helps,

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General advice: consider the integral test for convergence. As an exercise, apply it to the series $\sum\limits_{n = 2}^\infty {\frac{1}{{n(\log n)^\alpha }}}$, $\alpha > 0$ fixed. For which $\alpha$ does the series coverge?

Another very useful test is the alternating series test. See also convergence tests.

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Sorry to ruin your answer. We don't cover integral test in our course. :( – Qosmo Feb 17 '11 at 19:46