Introduction

Welcome to this guide on Representing Functions as Power Series, this article will guide you through the following:

Functions and Power Series

As one sees in the Taylor Series section, a function (like trigonometric functions) can be represented as an infinite series, for example:

$e^x=\sum _{n=0}^{\infty }\frac{x^n}{n!}$

$\sin x=\sum _{n=0}^{\infty }\frac{(-1{)}^n{x}^{2n+1}}{(2n+1)!}$

This hints to us that perhaps all functions can be written as an infinite series. 

Example 1:

1. Represent $f(x)=\int e^{-x^2}dx$ as a power series

This function cannot be obtained easily, since directly integrating $e^{-x^2}$ requires techniques that is not taught in Calc BC, however, recall that

$e^{-x^2}=\sum _{n=0}^{\infty }(-1{)}^{n+1}\frac{x^{2n}}{n!}=1-x^2+\frac{x^4}{2!}-\frac{x^6}{3!}+\frac{x^8}{4!}-\dots$

We can integrate this term by term, and obtain this function

$f(x)=\int e^{-x^2}dx=\int \sum _{n=0}^{\infty }(-1{)}^{n+1}\frac{x^{2n}}{n!}dx$

Notice that $(-1{)}^{n+1}$ and $n!$ are all constant, therefore we only need to worry about the $x$ term in the series.

$f(x)=\sum _{n=0}^{\infty }\frac{(-1{)}^{n+1}}{n!}\int x^{2n}dx=\sum _{n=0}^{\infty }(-1{)}^{n+1}\frac{x^{2n+1}}{(2n+1)n!}$

Just like functions can be represented as a power series, a power series can also be turned into a function.

Example 2:

Represent $f(x)=\sum _{n=0}^{\infty }3x^n$, where $\left|x\right|<1$

In the section on Taylor series, we derived that 

$\sum _{n=0}^{\infty }{x}^n=\frac{1}{1-x}$

Therefore

$f(x)=\sum _{n=0}^{\infty }3x^n=\frac{3}{1-x}$

Practice