Convergence of $\sum_{n=1}^\infty \frac{1}{n^\alpha \ln(1+n^\beta)}$

Question

For which values of $\alpha>0$ does the series $$\sum_{n=1}^\infty \frac{1}{n^\alpha \ln(1+n^\beta)}, \beta>1$$ converge?

Answer 1

Note that if $n \ge 2$ then $n^\beta < 1 + n^\beta < n^{2\beta}$ so that $$n \ge 2 \implies \beta \ln n < \ln(1 + n^\beta) < 2\beta \ln n.$$ By the comparison test, your series converges if and only if $$\sum_{n=1}^\infty \frac{1}{n^\alpha \ln n}$$ converges. Do you know how to proceed from here?

Answer 2

Use Cauchy condensation test to show that the given series converges iff $\sum {2^n a_{2^n}}$ converges,where $a_n$ is the general term of the series.