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The ratio and root test intuitively works by checking the long term behavior of a series $\sum a_n$ as a geometric series. In fact, that's the essence of the rigorous proof of the ratio test.

My question is what if instead of the geometric series, we use some other family of series that we use to check the long term behavior of a series for convergence? (maybe like $\sum \frac{1}{n^p}$)

zetapenguin
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  • The Cauchy condensation test converts a convergent/divergent $p$-series into a convergent/divergent geometric series, and converts a convergent/divergent "logarithmic $p$-series" to a convergent/divergent $p$-series (see here), and similarly for higher-order logarithmic $p$-series that are called generalized Bertrand series -- each application of the Cauchy condensation test removes one "logarithmic level" -- see here. – Dave L. Renfro Mar 06 '23 at 20:04
  • Here is a general criterion: https://en.m.wikipedia.org/wiki/Direct_comparison_test See also this (only in German...): https://de.m.wikipedia.org/wiki/Kriterium_von_Raabe – PhoemueX Mar 06 '23 at 21:15

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