Convergence of series with $\ln^3$

Question

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

Answer 1

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

Answer 2

Just as David pointed out:

$$\int\limits_1^\infty\frac1{(x+3)\log^3(x+3)}dx=\left.-\frac12\frac1{\log^2(x+3)}\right|_1^\infty=\frac12\frac1{\log^24}=\frac18\frac1{\log^22}$$

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

$$\frac{2^n}{(2^n+3)\log^3(2^n+3)}\le\frac1{n^3\log^32}$$

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