# Use the Integral Test to determine whether the series is convergent or divergent given sum 1 / n^5 from n=1 to infinity?

The Integral Test says that if ${a}_{n} > 0$ is a sequence and $f \left(x\right) > 0$ is monotonic decrescent and ${a}_{n} = f \left(n\right) \forall n$,
${\int}_{1}^{+ \infty} f \left(x\right) \mathrm{dx}$ exists finite $\iff {\sum}_{n = 1}^{+ \infty} {a}_{n}$ converges.
We know ${\int}_{1}^{+ \infty} \frac{1}{x} ^ 5 \mathrm{dx} = {\left(- \frac{1}{4 {x}^{4}}\right)}_{1}^{+ \infty} = \frac{1}{4}$, which is finite, so the series converges.