Power Series

By SkytrailGroup Mathematics Editorial Team April 12, 2026

Definition

\[\sum_{{n=0}}^\infty c_n(x-a)^n,\quad R=\frac{{1}}{{\limsup|c_n|^{{1/n}}}}\]

Examples

Example 1. Radius of convergence of \(\sum x^n/n\).

Solution. Ratio test: R=1.

In Depth

A power series \(\sum_{n=0}^\infty c_n(x-a)^n\) is an infinite polynomial centered at \(a\). Within its radius of convergence \(R\), it converges absolutely and defines a smooth (infinitely differentiable) function. Outside \(|x-a|>R\), it diverges.

The radius of convergence is determined by the Cauchy–Hadamard formula: \(1/R = \limsup_{n\to\infty} |c_n|^{1/n}\). Equivalently, the ratio test gives \(R = \lim |c_n/c_{n+1}|\) when this limit exists.

Power series can be differentiated and integrated term by term within the radius of convergence. This makes them powerful tools for solving differential equations (the power series method) and for computing integrals without closed forms.

Key power series: \(e^x = \sum x^n/n!\) (\(R=\infty\)), \(\sin x = \sum (-1)^n x^{2n+1}/(2n+1)!\) (\(R=\infty\)), \(\cos x = \sum (-1)^n x^{2n}/(2n)!\) (\(R=\infty\)), \(1/(1-x) = \sum x^n\) (\(R=1\)), \(\ln(1+x) = \sum (-1)^{n+1} x^n/n\) (\(R=1\)).

Key Properties & Applications

A power series \(\sum_{n=0}^\infty a_n(x-c)^n\) converges absolutely for \(|x-c|R\). The radius is \(R=1/\limsup|a_n|^{1/n}\) (Cauchy–Hadamard formula). Behavior at \(|x-c|=R\) requires separate analysis.

Within the radius of convergence, power series can be differentiated and integrated term by term. This makes them powerful: the Taylor series of \(e^x\), \(\sin x\), \(\cos x\), and \(\ln(1+x)\) are computed by differentiating and evaluating at \(x=0\).

Power series are used to define functions (\(e^z=\sum z^n/n!\) for complex \(z\)), solve differential equations (Frobenius method), and compute values numerically. The error in truncating at degree \(N\) is bounded by the first omitted term (for alternating series) or by Taylor's remainder theorem.