For which values of the parameter $p$ the following series is convergent?

Question

For which values of the parameter $p$ the following series is convergent? $$ \sum_{n=2}^{\infty} \frac{1}{n \ln^p n } $$

First, for $p=0$ , the series doesn't converge. For $p<0$ , the series also diverges, since: $\ln^p n < n$ which implies: $ \frac{1}{n \ln^p n } \geq \frac{n^{-p}}{n} $ and the series in the RHS diverges,

Will you please help me with the case $p>0$ ?

Thanks

To evaluate $\int{1\over x(\ln x)^p},dx$, substitute $u=\ln x$. — David Mitra, Apr 17 '15 at 07:57
http://math.stackexchange.com/questions/1237015/find-the-values-of-p-for-the-following-check-if-they-are-converges — Jesse P Francis, Apr 17 '15 at 08:04

score 1 · Answer 1 · answered Apr 17 '15 at 08:36

1

By Cauchy's condensation test, such a series is convergent for any $p>1$, divergent otherwise.

answered Apr 17 '15 at 08:36

Jack D'Aurizio

353,855

For which values of the parameter $p$ the following series is convergent?

1 Answers1