---
tag: 0VP56B4EMX
kind: definition
name: Power series
msc: [30B10]
defines: 0MZB06ST2B
uses: [0JH4MN4K1C]
sources:
  - url: https://en.wikipedia.org/wiki/Power_series
    locator: lead section
---

## Definition

Let $c$ be a real number and let $(a_n)_{n \ge 0}$ be a real sequence.
The power series with center $c$ and coefficients $a_n$ is the [[series|0JH4MN4K1C]]
$$\sum_{n=0}^{\infty} a_n (x - c)^n,$$
taken as a function of the real variable $x$.

## Remarks

The series always converges at $x = c$, with sum $a_0$.
The set of $x$ where it converges is an interval around $c$. Its half length is the [[radius of convergence|0660QX1VPR]].
Polynomials are power series with only finitely many nonzero coefficients.
Complex power series use the same definition, with a complex variable $z$.

## Explanations

### L1 human

A power series is a polynomial with infinitely many terms.
For each $x$, we put $x$ into every term and add up the resulting series of numbers.