# Integral Test for Convergence of an Infinite Series

## Key Questions

• Well, I would avoid using the integral test since evaluating an integral can be very difficult. If nothing else works and you know how to evaluate the integral, then go for it.

• By Integral Test,

${\sum}_{n = 1}^{\infty} \frac{1}{n} ^ 5$ converges.

Let us look at some details.

Let us evaluate the corresponding improper integral.

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

$= {\lim}_{t \to \infty} {\int}_{1}^{t} {x}^{- 5} \mathrm{dx}$

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

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

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

$= - \frac{1}{4} \left(0 - 1\right) = \frac{1}{4}$

Since the integral

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

converges to $\frac{1}{4}$,

${\sum}_{n = 1}^{\infty} \frac{1}{n} ^ 5$

also converges by Integral Test.

• Integral Test

If $f$ is a function such that $f \left(n\right) = {a}_{n}$, then

${\sum}_{n = 1}^{\infty} {a}_{n}$ and ${\int}_{1}^{\infty} f \left(x\right) \mathrm{dx}$ converge or diverge together.

I hope that this was helpful.