Which convergence test should be used for this series? The series is, $$\sum \limits_{n = 1}^\infty \frac{ \ln(1 + \frac{5}{n})}{\sqrt [3] {n+1}}.$$ Using the asymptotic equivalence concept I came up with, $$\frac{ \ln(1 + \frac{5}{n})}{\sqrt [3] {n+1}} \sim \frac{ \ln(\frac{1}{n})}{\sqrt [3]{n}} (n \to \infty)$$ But now, I can't figure out what test to use. I have tried the comparation test and the limit comparation test and I couldn't do it. Can you give me a hint? Thanks.

You used wrongly the asymptotic equivalence: recall that $$\log(1+x)\sim_0 x$$ so we have $$\frac{ \ln(1 + \frac{5}{n})}{\sqrt [3] {n+1}}\sim_\infty \frac{5}{n\sqrt [3] {n}}=\frac{5}{n^{4/3}}$$ and the Riemann series $\displaystyle \sum_{n\ge1}\frac{5}{n^{4/3}}$ is convergent so by asymptotic comparison the given series is convergent.