Mathematical Methods in the Physical Sciences · Problem 6.15

INFINITE SERIES, POWER SERIES — Mary Boas

Mathematics — Methods · integral-test

Problem

Use the integral test to prove the following so-called

p

-series test. The series

\sum_{n=1}^{\infty} \frac{1}{n^p} \text{ is } \begin{cases} \text{convergent if } p > 1, \\ \text{divergent if } p \leq 1. \end{cases}

Caution: Do

p = 1

separately.