Power Series

Overview

A power series in $z - a$ is a complex-valued series of form

\sum_{n = 0}^{\infty} a_{n} (z - a)^{n} .

The numbers $a_{0}, a_{1}, a_{2}, \dots \in C$ are called coefficients of the power series. The number $a \in C$ is called the center of the power series. Every power series is associated with a circle of convergence centered at $a$ . The radius of this circle is called the radius of convergence $R$ .

The series absolutely converges for every $z$ with $| z | < R$ . It also uniformly converges on every closed disk with center at $a$ and radius less than $R$ . The series diverges for every $z$ with $| z | > R$ . Points on the boundary are indeterminate.

Real Expansion

Each real power series defines a sum function whose value at each $x$ in the interval of convergence is given by

f (x) = \sum_{n = 0}^{\infty} a_{n} (x - a)^{n} .

The series is said to represent the function $f$ in the interval of convergence, and it is called the power-series expansion of $f$ about $a$ .

Differentiation

Let $f$ be represented by the following real power series in $(a - r, a + r)$ :

f (x) = \sum_{n = 0}^{\infty} a_{n} (x - a)^{n}

Then the derivative $f^{'} (x)$ exists for each $x$ in the interval of convergence and is given by

f^{'} (x) = \sum_{n = 1}^{\infty} n a_{n} (x - a)^{n - 1} .

The radius of $f^{'} (x)$ 's power-series expansion's interval of convergence is the same as that of $f (x)$ .

Uniqueness

If two real power series

\sum a_{n} (x - a)^{n} and \sum b_{n} (x - a)^{n}

have the same function $f$ in the same neighborhood of the point $a$ , then the two series are equal term by term. In particular, for each $n \geq 0$ ,

a_{n} = b_{n} = \frac{f^{(n)} (a)}{n!} .

This shows the partial sums of a power series represented by $f$ at $a$ are the Taylor polynomials of degree $n$ generated by $f$ at $a$ .

Taylor Series

Let $f$ be a function with derivatives of every order in an open interval about $a$ . Then the Taylor series generated by $f$ at $a$ is the power series

\sum_{k = 0}^{\infty} \frac{f^{(k)} (a)}{k!} (x - a)^{k} .

Though such a Taylor series can be formed, the corresponding sum function may not coincide with $f$ . It will coincide if and only if the remainder $E_{n} (x) \to 0$ as $n \to \infty$ .

A Taylor series is also called a Maclaurin series when centered about point $0$ .

Special Cases

Assume complex $x$ . The following Maclaurin series frequently occur in analysis:

\begin{aligned} e^{x} & = \sum_{n = 0}^{\infty} \frac{x^{n}}{n!} & for all x \\ \frac{1}{1 - x} & = \sum_{n = 0}^{\infty} x^{n} & for | x | < 1 \\ \ln (1 - x) & = - \sum_{n = 1}^{\infty} \frac{x^{n}}{n} & for | x | < 1 \land x = - 1 \\ \ln (1 + x) & = \sum_{n = 1}^{\infty} (- 1)^{n + 1} \frac{x^{n}}{n} & for | x | < 1 \land x = 1 \\ \sin x & = \sum_{n = 0}^{\infty} \frac{(- 1)^{n}}{(2 n + 1)!} x^{2 n + 1} & for all x \\ \cos x & = \sum_{n = 0}^{\infty} \frac{(- 1)^{n}}{(2 n)!} x^{2 n} & for all x \\ \arctan x & = \sum_{n = 0}^{\infty} \frac{(- 1)^{n}}{2 n + 1} x^{2 n + 1} & for | x | \leq 1 \end{aligned}