Mathematical Methods in the Physical Sciences · Problem 6.30

INFINITE SERIES, POWER SERIES — Mary Boas

Mathematics — Methods · ratio-test

Problem

Prove the ratio test. Hint: If

|a_{n+1}/a_n| \to \rho < 1

, take

\sigma

so that

\rho < \sigma < 1

. Then

|a_{n+1}/a_n| < \sigma

n

is large, say

n \geq N

. This means that we have

|a_{N+1}| < \sigma|a_N|

|a_{N+2}| < \sigma|a_{N+1}| < \sigma^2|a_N|

, and so on. Compare with the geometric series

\displaystyle\sum_{n=1}^{\infty} \sigma^n|a_N|

. Also prove that a series with

\rho > 1

diverges. Hint: Take

\rho > \sigma > 1

, and use the preliminary test.