# Math Help - Series Question

1. ## Series Question

$
\sum\limits_{n = 1}^\infty {\frac{{5\ln n}}
{{\sqrt {n^3 } }}}$

Is there any easy way to find the convergence/divergence of this series?

So far I tried using the integral test and came up with a convergence at 20.
This involves integration by substitution. Is there any easier method?

2. Hello,
Originally Posted by RedBarchetta
Is there any easier method?
Maybe : $\sqrt{n^3}=n^\frac32=n^\frac54n^\frac14$ hence

$
5\sum_{n = 1}^\infty\frac{\ln n}{\sqrt {n^3 }}=5\sum_{n = 1}^\infty\frac{1}{n^\frac54}\times\frac{\ln n}{n^\frac14}$
.

As $\frac{\ln n}{n^\frac14}\xrightarrow[n\to\infty]{}0$ one has, for $n$ sufficiently large, $\left|\frac{\ln n}{n^\frac14}\right|\leq 1$ hence $\left|\frac{1}{n^\frac54}\times\frac{\ln n}{n^\frac14}\right|\leq\left|\frac{1}{n^\frac54}\ right|\times 1 = \left|\frac{1}{n^\frac54}\right|$ that is to say $\left|\frac{\ln n}{\sqrt {n^3 }}\right|\leq\left|\frac{1}{n^\frac54}\right|$. As $\sum\frac{1}{n^\frac54}$ is an absolutely convergent series, the series $5\sum_{n = 1}^\infty\frac{\ln n}{\sqrt {n^3 }}$ converges too. (comparison test)

3. Thanks.