Integral Test

Summary

For a positive, continuous, eventually decreasing function $f$ , the series $\sum f(n)$ and the improper integral ${\int }_1^{\mathrm{\infty }}f(x)dx$ either both converge or both diverge.

Prerequisites

Integrals, improper integrals, Sequences

Theorem

Let $f$ be positive, continuous, and decreasing on $[N,\mathrm{\infty })$ for some integer $N\ge 1$ . Then

Worked Example

For $f(x)=x^{-p}$ ( $p>0$ ):

converges if and only if $p>1$ . Thus $\sum 1/n^p$ converges iff $p>1$ (see P Series).

For $\sum 1/\sqrt{n}$ , ${\int }_1^{\mathrm{\infty }}{x}^{-1/2}dx=\mathrm{\infty }$ , so the series diverges.

Common Mistakes

• Using the integral’s value as the sum of the series (they are not equal; the integral only decides convergence and bounds remainders).
• Applying the test when $f$ is not eventually decreasing.

Connections

• P Series, Harmonic Series, remainder bounds in Estimating Series Sums

References

The integral test is in OpenStax Calculus Volume 2.^[1]