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So after taking calculus II, or maybe a first course in analysis, everyone learns a few series tests. They learn

1) Divergence Tests

2) Integral Test (from which we deduce things like $p$-series.

3) Comparison Tests

4) Root and Ratio Tests

As well as a few useful theorems pertaining to geometric series, telescoping series, a series-analog to the squeeze theorem. Maybe a few others, but that's all I can come up with off the top of my head for an introductory course. But what other series tests are out there?

I know that there are two tests which are associated, one by Raabe and one by Gauss. Another one called Dirichlet's Test. I get the feeling there are more than just these. Can anyone throw out some others for series of a single variable?

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  • $\begingroup$ What did you find when you typed "series convergence" into Google? $\endgroup$ Oct 21, 2014 at 6:22
  • $\begingroup$ You did not mention en.m.wikipedia.org/wiki/Cauchy_condensation_test. $\endgroup$
    – PhoemueX
    Oct 21, 2014 at 6:42
  • $\begingroup$ I found calculus II references everywhere. That's why I'm asking here. When I looked for other series, I came across papers and other writings above my level. $\endgroup$ Oct 21, 2014 at 13:12

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Here are some references. The bibliographies in these will lead you to many more references. Based on your comments, Bonar/Khoury's book is probably the best fit for you right now.

Bonar/Khoury, Real Infinite Series (2006)

Bromwich, An Introduction to the Theory of Infinite Series (1908) [a 1925 2nd edition also exists]

Hirschman, Infinite Series (1962/2014)

Knopp, Theory and Application of Infinite Series (1954)

Knopp, Infinite Sequences and Series (1956)

Osgood, Introduction to Infinite Series (1897)

Rainville, Infinite Series (1967)

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