 # Power Series – Definition, General Form, and Examples

The power series is one of the most useful types of series in mathematical analysis.  We can use power series to define different transcendental functions including the exponential and trigonometric functions. Understanding the power series will also make us appreciate how we have approximated functions’ values in our calculators and computers.

The power series allows us to approximate functions as the sum of the powers of the variable. We can think of power series as an infinite polynomial that leads to the approximation of common and new functions.

In this article, we’ll explore the definition of the power series and learn how to define common and new functions through this expansion. We’ll also show you how to confirm whether the given power series is convergent or divergent.

In order for us to do so, make sure to refresh your knowledge on the following:

For now, let’s begin by understanding the unique components of a power series.

## What is a power series?

The power series, centered at $c$, is a series represented by the general form shown below.

\begin{aligned}\sum_{n = 0}^{\infty} a_n(x -c)^n &= a_0 + a_1(x -c) + a_2( x- c)^2 + …\end{aligned}

The constants $\boldsymbol{a_n}$, where $n \geq 0$, are called the series’ coefficients and $\boldsymbol{c}$ represents the center. Notice something unique about this series as opposed to the ones we’ve learned in the past? The terms of the power series are now all in terms of $x$ when in the past, we’ve only worked with series that contains numbers. This will be the start of our understanding of series and expansions of common and new functions.

 Examples of Power Series Expansion\begin{aligned}\sin x &= x – \dfrac{x^3}{3!} + \dfrac{x^5}{5!} – \dfrac{x^7}{7!}+ …\\\cos x &= 1 – \dfrac{x^2}{2!} + \dfrac{x^4}{4!} – \dfrac{x^6}{6!}+ …\end{aligned}

Here are two great examples of a power series- the power series of $\sin x$ and $\cos x$. Through this amazing series, we can now express transcendental functions such as sine and cosine functions as a series of polynomials. In the next sections, we’ll learn how to apply the power series formula and understand the process of expressing functions as a power series.

## What is the power series formula?

We’ll show you two variants of the power series formula: 1) expression, when $\boldsymbol{x}$ is centered at zero and 2,) when $\boldsymbol{x}$ is centered at $\boldsymbol{c}$.

Suppose that $\{a_n\}$ is a sequence, $x$ is the variable, and $c$ represents a real number. We have the following power series formula:

Power series centered at the origin:

\begin{aligned}\boxed{\boldsymbol{\sum_{n = 0}^{\infty} a_nx^n = a_0 + a_1x + a_2x^2 + a_3x^3+…}}\end{aligned}

Power series centered at variable $\boldsymbol{c}$:

\begin{aligned}\boxed{\boldsymbol{\sum_{n = 0}^{\infty} a_n(x – c)^n = a_0 + a_1(x – c) + a_2(x – c)^2 + a_3(x – c)^3+…}}\end{aligned}

For a more precise definition, we establish that $x^0 = 1$ and $(x – c)^0 = 1$ even when $x = 0$ and $x = c$, respectively.

The simplest example of a power series and this occurs when $x$ is centered at the origin and when $a_n = 1$.

\begin{aligned}f(x) &= \sum_{n = 0}^{\infty} x^n\\&= 1 + x + x^2 + x^3 + …,\end{aligned}