University Maths Notes: Analysis – Radius of Convergence of Power Series

Not all sequences either always converge or always diverge. Often a series converges for some values ofand diverges for other values of

If a function is expanded in a power series about a pointso thatthen the set of values ofaboutfor which the series converges is called the interval of convergence and if the series converges for alland diverges for allthenis the radius of convergence.

The interval of convergence can often be found using the ratio test with the conditionor

Example: find the interval of convergence for the power series

Cancellation givesAllowing now shows that interval of convergence is

Example: find the interval of convergence for the power series

Cancellation givesLet then

The radius of convergence isand the interval of convergence is