# How do you use the Integral test on the infinite series sum_(n=1)^oo1/root5(n) ?

Sep 17, 2014

Since the integral
${\int}_{1}^{\infty} \frac{1}{\sqrt[5]{x}} \mathrm{dx}$
diverges, we can conclude that the series
${\sum}_{n = 1}^{\infty} \frac{1}{\sqrt[5]{n}}$
also diverges by Integral Test.

Here is the details of the evaluation of the integral.

${\int}_{1}^{\infty} \frac{1}{\sqrt[5]{x}} \mathrm{dx}$

$= {\lim}_{t \to \infty} {\left[\frac{5}{4} {x}^{\frac{4}{5}}\right]}_{1}^{t}$

$= \frac{5}{4} {\lim}_{t \to \infty} \left({t}^{\frac{4}{5}} - 1\right)$

$= \frac{5}{4} \left(\infty - 1\right) = \infty$