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?

  • $\begingroup$ 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

1 Answer 1


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)


You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .