Answer
Convergent
Work Step by Step
Given: $f(x)=x^{-5}$.
Since $f$ is continuous, positive, and decreasing for $x\gt 1$, we can use the Integral Test to determine whether the series is convergent or divergent.
We have
$\int_1^\infty f(x)dx=\int_1^\infty x^{-5}dx=\lim\limits_{t \to \infty}\int_1^tx^{-5}dx=\lim\limits_{t \to \infty}\left[\frac{x^{-4}}{-4}\right]_1^t=\lim\limits_{t \to \infty}[-\frac{1}{4t^{4}}+\frac{1}{4}]$
$=0+\frac{1}{4}$
$=\frac{1}{4}$
Hence, the given series converges.