A power series is a series of the form $$\sum_{k=0}^{\infty}a_k(x-x_0)^k,$$ with $a_k,x_0\in\mathbb{R}$ or $\in\mathbb{C}$ . The $a_k$ are called the coefficients and $x_0$ the center of the power series. $a_0$ is called the constant term.

Where it converges the power series defines a function, which can thus be represented by a power series. This is what power series are usually used for. Every power series is convergent at least at $x=x_0$ where it converges to $a_0$ . In addition it is absolutely and uniformly convergent in the region $\{x\mid |x-x_0|<r\}$ , with $$r=\liminf_{k\to\infty}\frac{1}{\sqrt[k]{|a_k|}}$$ It is divergent for every $x$ with $|x-x_0| > r$ . For $|x-x_0|= r$ no general predictions can be made. If $r=\infty$ , the power series converges absolutely and uniformly for every real or complex $x.$ The real number $r$ is called the radius of convergence of the power series.

Examples of power series are:

• Taylor series, for example: $$e^x=\sum_{k=0}^{\infty}\frac{x^k}{k!}.$$
• The geometric series: $$\frac{1}{1-x}=\sum_{k=0}^{\infty}x^k,$$ with $|x|<1$ .

Power series have some important properties:

• If a power series converges for a $z_0\in\mathbb{C}$ then it also converges for all $z\in\mathbb{C}$ with $|z-x_0|<|z_0-x_0|$ .
• Also, if a power series diverges for some $z_0\in\mathbb{C}$ then it diverges for all $z\in\mathbb{C}$ with $|z-x_0|>|z_0-x_0|$ .
• For $|x-x_0|<r$ Power series can be added by adding coefficients and multiplied in the obvious way: $$\sum_{k=0}^\infty a_k(x-x_o)^k\cdot\sum_{l=0}^\infty b_j(x-x_0)^j = a_0b_0+(a_0b_1+a_1b_0)(x-x_0)+(a_0b_2+a_1b_1+a_2b_0)(x-x_0)^2\ldots.$$
• (Uniqueness) If two power series are equal and their centers are the same, then their coefficients must be equal.
• Power series can be termwise differentiated and integrated. These operations keep the radius of convergence.