Does $\displaystyle\sum{\ln n\over n^{4/3}}$ converge or diverge?
Which test should I use?
I tried the ratio test and root test but both of them are inconclusive.
I could try the comparison test but I don't know which function I have to compare with.
[Math] Does $\sum{\ln n\over n^{4/3}}$ converge or diverge
calculusconvergence-divergencedivergent-seriesintegrationsequences-and-series
Best Answer
By the integral test you may compare your series with the following integral: $$ \int_1^\infty \frac{\ln x}{x^{4/3}} dx=\left.\frac{x^{-4/3+1}}{-4/3+1}\ln x\right|_1^\infty-\frac1{-4/3+1}\int_1^\infty x^{-4/3+1}\frac1{x} dx=9<\infty $$ giving the convergence of your series $$ \sum_1^\infty\frac{\ln n}{n^{4/3}}. $$