Title: 16_4 Power Series
Course: HELM Workbooks 2023-2024 > Section: HELM 16: Sequences and Series

In this Section we consider power series. These are examples of infinite series where each term contains a variable, \(x\), raised to a positive integer power, that is \(\displaystyle{\sum_{n=0}^\infty a_nx^n}\) where \((a_n)_{n\geq 0}\) is an infinite series of real numbers.

We use the ratio test to obtain the radius of convergence, \(R\), of the power series, this is the result that the series is absolutely convergent if \(|x|<R\), divergent if \(|x|>R\), and may or may not be convergent if \(x=\pm R\).

Finally, we extend the work to apply to general power series when the variable \(x\) is replaced by \((x − x_0)\), that is, series of the form \(\displaystyle{\sum_{n=0}^\infty a_n(x-x_0)^n}\). Note that here we say the series is 'centered' at \(x_0\)