One of the most useful tests for the convergence of series is the p-series test. It is a comparison test that is deeply related to the convergence of a class of improper Riemann integrals.

P-series Test proof

To prove the p-series test, we need some preliminaries on the relationship between improper integrals and the series.

Integral test for series

Theorem: Let $f:[0,+\infty)$ be a positive continuous decreasing function. Then for any $k\in\mathbb{N},$ the following assertions hold

  • If the improper integral $\displaystyle\int^{+\infty}_k f(x)dx$ converges, then the series $\displaystyle\sum_{n=k}^{+\infty} f(n)$ converges.
  • If the improper integral $\displaystyle\int^{+\infty}_k f(x)dx$ diverges, then the series $\displaystyle\sum_{n=k}^{+\infty} f(n)$ diverges.