How to prove that $\sum \frac {1}{(n+3)\ln^ 3 (n+3)} $ converges?


Just as David pointed out:


and thus our series is convergente, or by the Condensation Test:


and since the series with general term $\;n^{-3}\;$ converges so does ours.

  • $\begingroup$ I think as in the title, the OP wanted the integral version. +1 $\endgroup$ – mrs Dec 24 '13 at 18:08

The Bertrand series $$\sum_{n\ge2}\frac{1}{n^\alpha\ln^\beta n}$$ is convergent if and only if $(\alpha>1)\lor(\alpha=1\land\beta>1).$


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