Answer
Convergent
Work Step by Step
$\sum_{n=1}^{\infty} \frac{1}{(5n-1)^4}$
Before applying the Intergal Test, we need to ensure that $f(n)$ is decreasing, let
$f(x)=\frac{1}{(5x-1)^4}\implies f'(x)=-\frac{1}{20}\frac{1}{(5x-1)^5}$
Above equation implies that $f(x)$ is decreasing if $x>\frac{1}{5}$
Now by apply theIntegral Test, we have:
$\int_{1}^{\infty}\frac{1}{(5x-1)^4}=\lim_{t\to\infty}\int_{1}^{t}\frac{1}{(5x-1)^4}=-\frac{1}{15}\lim_{t\to\infty}\biggl[\frac{1}{(5x-1)^3}\biggr]_{1}^{t}=\frac{1}{960}$
Since the integral is convergent, the series is also convergent by Integral Test.
