Power Series

Level 1 - Calculus topic page in Series And Approximations.

Principle

A power series is a series of powers of \(x-a\). It behaves like an infinite polynomial within its interval of convergence.

\(\sum c_n(x-a)^n\)

power series centred at a

\(R\)

radius of convergence

\(c_n\)

coefficient of the nth power

Use a convergence test, often the ratio test, to find values of \(x\) for which the series converges.

Power series form

\[\sum_{n=0}^{\infty}c_n(x-a)^n\]

The result is usually \(|x-a|<R\), plus separate endpoint tests. Endpoint behaviour is not decided by the radius alone.

Question

Find the radius of convergence of \(\sum_{n=0}^{\infty}x^n\).

Answer

This is a geometric series with ratio \(x\). It converges when \(|x|<1\). Therefore the radius of convergence is \(R=1\).

Question

For \(\sum_{n=0}^{\infty}(x-2)^n\), state the centre and radius.

Answer

The series is geometric with ratio \(x-2\), so it converges when \(|x-2|<1\). The centre is \(2\), and the radius is \(1\).