6.4 Working with taylor series

In the preceding section, we defined Taylor series and showed how to find the Taylor series for several common functions by explicitly calculating the coefficients of the Taylor polynomials. In this section we show how to use those Taylor series to derive Taylor series for other functions. We then present two common applications of power series. First, we show how power series can be used to solve differential equations. Second, we show how power series can be used to evaluate integrals when the antiderivative of the integrand cannot be expressed in terms of elementary functions. In one example, we consider ${\displaystyle \int e^{\text{−}{x}^2}}dx,$ an integral that arises frequently in probability theory.

The binomial series

Our first goal in this section is to determine the Maclaurin series for the function $f\left(x\right)={\left(1+x\right)}^r$ for all real numbers $r.$ The Maclaurin series for this function is known as the binomial series    . We begin by considering the simplest case: $r$ is a nonnegative integer. We recall that, for $r=0,1,2,3,4,f\left(x\right)={\left(1+x\right)}^r$ can be written as

The expressions on the right-hand side are known as binomial expansions and the coefficients are known as binomial coefficients. More generally, for any nonnegative integer $r,$ the binomial coefficient of $x^n$ in the binomial expansion of ${\left(1+x\right)}^r$ is given by

and

For example, using this formula for $r=5,$ we see that

We now consider the case when the exponent $r$ is any real number, not necessarily a nonnegative integer. If $r$ is not a nonnegative integer, then $f\left(x\right)={\left(1+x\right)}^r$ cannot be written as a finite polynomial. However, we can find a power series for $f.$ Specifically, we look for the Maclaurin series for $f.$ To do this, we find the derivatives of $f$ and evaluate them at $x=0.$

We conclude that the coefficients in the binomial series are given by

We note that if $r$ is a nonnegative integer, then the $\left(r+1\right)\text{st}$ derivative $f^{\left(r+1\right)}$ is the zero function, and the series terminates. In addition, if $r$ is a nonnegative integer, then [link] for the coefficients agrees with [link] for the coefficients, and the formula for the binomial series agrees with [link] for the finite binomial expansion. More generally, to denote the binomial coefficients for any real number $r,$ we define

With this notation, we can write the binomial series for ${\left(1+x\right)}^r$ as