Why Do We Care?

If we can define a function with a power series it can be much easier to work with the function (especially if we only want approximate values of the function). Why? Because polynomials are nice functions to work with. We can always find an approximate for a function by taking a finite number of terms from the infinite series. As long as we are close to the value that we are centered at we will be able to make good approximations. For example, consider the function $\sin(x)$ and the first 4 terms from its power series centered about $x=0$.

As long as we are close to zero the function $f(x)=x-x^3/3!+x^5/5!-x^7/7!$ is a good approximation for $\sin(x)$. So if we wanted to know the value of $\sin(.1)$ we could just evaluate $f(.1)$ and have a good approximation. If we are not accurate enough we would simply go out more terms in the series.

We also used power series to solve some differential equations. We solved $y'=xy$ earlier in the semster. Just by making the assumption that we could write the function $y(x)$ as a polynomial we were able to determine the coefficients of that polynomial with some simple algebra. We will also use a power series to approximate $\pi$. This is a famous problem in mathematics, determining $\pi$ to as many digits as possible. Some have even found $\pi$ to over 2 billion digits. The techniques relied on power series.

Power series should be thought of as the elementary definitions of functions. For example, the power series for $f(x)=\cos\sqrt{x}$ is

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

If we interpret this function geometrically then it makes no sense for $x<0$. What is the cosine of an imaginary number? But if we realize that $\cos\sqrt{x}$ is really its power series (over the interval of convergence of course), then we can make sense of $\cos(i)$ where $i^2=-1$. We simply evaluate the power series at $x=-1$.

$$\cos(\sqrt{-1})=\displaystyle \sum^{\infty}_{n=0}(-1)^n\displayst... ...{(-1)^n}{(2n)!}= \displaystyle \sum^{\infty}_{n=0}\displaystyle \frac{1}{(2n)!}$$

There are applications in digital processing where certain amplitudes are functions of complex numbers. So we need to be able to evaluate these periodic functions over the complex numbers as well as the real numbers. Power series are clearly one of the most useful tools one can learn about in a Calculus course.