Integral test

Statement

Let $N$ be an integer and let $f \colon [N, \infty) \to \mathbb{R}$ be a decreasing function with $f(x) \ge 0$ for all $x \ge N$. Then the series $\sum_{n=N}^{\infty} f(n)$ converges if and only if the improper integral $\int_N^{\infty} f(x)\, dx$ converges.

Remarks

A decreasing function is Riemann integrable on each bounded interval, so the integrals $\int_N^b f(x)\, dx$ exist. The proof uses the comparison of a series and an integral. When both converge, $$\int_N^{\infty} f(x)\, dx \le \sum_{n=N}^{\infty} f(n) \le f(N) + \int_N^{\infty} f(x)\, dx.$$ With $f(x) = 1/x^p$ it gives the p-series test. Mathlib states the convergence direction for an antitone function, in AntitoneOn.summable_of_integrableOn_Ioi.

Explanations

L1 human

Draw rectangles of width $1$ and height $f(n)$ under or over the graph of $f$. The total area of the rectangles is the sum of the series, and it stays within $f(N)$ of the area under the graph.