Theory

Introduction

Polynomial functions have the form

$$p_k(x)=a_0+a_{1}x+a_2{x}^2+\cdots +a_k{x}^k.$$

A power series is an infinite sum of power terms

$$p(x)=a_0+a_{1}x+a_2{x}^2+a_3{x}^3+\dots =\sum _{n=0}^{\infty }{a}_n{x}^n$$

which is determined by an infinite series of coefficient numbers $a_0,a_1,a_2,a_3\dots$. It is an infinite series all of whose partial sums are polynomial functions.

If a power series is given with either:

• (1) the general term $a_n$ as a function of $n$ in terms of an algebraic formula,
• (2) the $n^{\text{th}}$ term $a_n$ as a function of all previous terms in a recursion relation,

then a computer can quickly produce the coefficients of a power series to any degree $N$, and approximately evaluate the series at any number of points $x$ by taking the required products and sums of numbers, up to the $N^{\text{th}}$ term.

Polynomials do approximate everything

Any continuous function $f(x)$ on an interval $[a,b]$ can be approximated by a sequence of polynomials with increasing degree.

However, such approximating polynomials might not form a power series, meaning that their initial coefficients do not get locked in, or in other words, the sequence of polynomials have ever differing initial coefficients.

Power series can not approximate everything

Many important functions in mathematics cannot be approximated by power series.

When a function is approximated by a power series, it is called analytic or (to distinguish from the complex case) real analytic.

Introduction (part 2)

One perspective on power series is the approximation perspective, according to which our goal is to approximate a class of potentially relevant functions using some standardized functions, and power series give a powerful type of such standardized functions. Even though many important functions are non-analytic, many other important ones are indeed analytic.

Another perspective on power series is the transcendental functions perspective, according to which our goal is to understand certain functions that are given to us and studied in the form of converging power series. This perspective takes an interest in higher transcendental functions. These functions are usually determined as power series in terms of an integral, an ODE (a more complex integral), or an asymptotic expansion of some kind. The so-called “special functions” like Bessel functions and Legendre functions and Chebyshev functions are of this kind.

Review of power series

A power series of the form

$$p(x)=a_0+a_{1}x+a_2{x}^2+\dots =\sum _{n=0}^{\infty }{a}_n{x}^n$$

is said to be expanded around $x=0$. A power series of the form

$$p(x)=a_0+a_1(x-c)+a_2(x-c{)}^2+\dots =\sum _{n=0}^{\infty }{a}_n(x-c{)}^n$$

is said to be expanded around $x=c$.

Radius of convergence
Power series have the interesting property of converging inside intervals centered at the expansion point $c$ (for real numbers), and disks centered at the expansion point $c$ (for complex numbers), and diverging outside these intervals. The distance from $c$ to the edge of the convergence domain is called the radius of convergence $R$.

Radius of convergence

The reason that power series converge inside intervals / disks that are symmetric about the expansion point $c$ is the root test, which is really a “comparison” to the geometric series $1+x+x^2+x^3+\dots$ that converges for $|x|<1$ and diverges for $|x|>1$. Inside the disk of convergence $|x-c|<R$, the quantities

$$\sqrt[n]{|a_n(x-c{)}^n|}=\sqrt[n]{|a_n|}\cdot |x-c|$$

are (eventually) all smaller than one, so the $n^{\text{th}}$ term of the power series is (eventually) smaller in absolute value than the $n^{\text{th}}$ term $r^n$ of a converging geometric series $1+r+r^2+\dots$, namely with $r<1$. Outside the disk of convergence, $|x-c|>R$, the individual terms of the series do not even converge to zero, so the series itself surely cannot converge.

Of course, it is important to note that the behavior of the series on the boundary may be non-symmetric. For example, $\sum \frac{x^j}{j}$ converges on $[-1,1)$ and diverges outside it. Remember, also, that the radius of convergence $R$ might be $\infty$.

Power series define functions on their domains of convergence. These functions are fairly convenient to handle using algebra and calculus.
Algebra
Power series can be added, subtracted, and multiplied:

$$\begin{array}{rl}\sum _{i=0}^{\infty }{a}_i{x}^i\pm \sum _{i=0}^{\infty }{b}_i{x}^i& =\sum _{i=0}^{\infty }(a_i\pm b_i)x^i,\\ \left(\sum _{i=0}^{\infty }{a}_i{x}^i\right)\cdot \left(\sum _{i=0}^{\infty }{b}_i{x}^i\right)& =\sum _{n=0}^{\infty }{c}_n{x}^n\end{array}$$

where $c_n=a_0{b}_n+a_1{b}_{n-1}+\cdots +a_n{b}_0$.

Calculus
It is easy to differentiate and integrate power series functions. This is one of their primary advantages.
Derivatives:

$$\begin{array}{rl}\frac{d}{dx}\sum _{n=0}^{\infty }{a}_n{x}^n& =\sum _{n=1}^{\infty }n{a}_n{x}^{n-1}\\ \frac{d^2}{d{x}^2}\sum _{n=0}^{\infty }{a}_n{x}^n& =\sum _{n=2}^{\infty }n(n-1)a_n{x}^{n-2}\\ \frac{d^3}{d{x}^3}\sum _{n=0}^{\infty }{a}_n{x}^n& =\sum _{n=3}^{\infty }n(n-1)(n-2)a_n{x}^{n-3}\\ & \dots .\end{array}$$

Integrals:

$$\int \sum _{n=0}^{\infty }{a}_n{x}^{n}dx=\sum _{n=0}^{\infty }{a}_n\frac{x^{n+1}}{n+1}+C.$$

The coefficients of a power series function can be extracted by evaluating the function and its derivatives:

$$a_0=f(0),a_1=f^{\prime }(0),a_2=f^{\prime \prime }(0),\dots ,a_n=f^{(n)}(0).$$

This fact entails that the coefficients of a power series are unique, and so any two power series with at least one differing coefficient will necessarily have differing values somewhere.

Power series have intrinsic merit, not just of other functions

We treat power series as functions on their interval of convergence.

Some functions are not necessarily initially defined using power series, and yet they “have” power series, for example $\sin (x)$ and $e^x$. This means that they are equal to a power series on some domain (possibly everywhere). This is an equality of two definitions.

For the most part, it is good to think of such a function as simply (defined by) its power series, and its other definition (involving geometry or integration or whatever) is actually a property of said power series, not a definition of said function.

Familiar examples

$$\begin{array}{rlrl}\frac{1}{1-x}& =\sum _{n=0}^{\infty }{x}^n& & =1+x+x^2+x^3+x^4+\cdots \\ e^x& =\sum _{n=0}^{\infty }\frac{x^n}{n!}& & =1+\frac{x^1}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\cdots \end{array}$$ $$\begin{array}{rlrl}\sin (x)& =\sum _{n=0}^{\infty }(-1{)}^n\frac{x^{2n+1}}{(2n+1)!}& & =\frac{x^1}{1!}-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots \\ \sinh (x)& =\sum _{n=0}^{\infty }\frac{x^{2n+1}}{(2n+1)!}& & =\frac{x^1}{1!}+\frac{x^3}{3!}+\frac{x^5}{5!}+\cdots \end{array}$$ $$\begin{array}{rlrl}\cos (x)& =\sum _{n=0}^{\infty }(-1{)}^n\frac{x^{2n}}{(2n)!}& & =1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\cdots \\ \cosh (x)& =\sum _{n=0}^{\infty }\frac{x^{2n}}{(2n)!}& & =1+\frac{x^2}{2!}+\frac{x^4}{4!}+\frac{x^6}{6!}+\cdots \end{array}$$

Power series solutions - basic technique

The fundamental idea is to convert an ODE into an algebraic system relating the coefficients of a power series function that solves the ODE, and then to solve this system to produce formulas for the coefficients.

Second-order linear equations

A number of famous special transcendental functions are determined by second-order differential equations

$$y^{\prime \prime }+p(x)y^{\prime }+q(x)y=0,$$

having the property that $p(x)$ and $q(x)$ are both analytic functions, meaning they are given by power series, or else are almost analytic (see the next section). A point $x_0$ is called an ordinary point of this ODE when $p(x)$ and $q(x)$ are indeed analytic in the vicinity of $x_0$. We treat ordinary points in this section.

Second-order linear equations (part 2)

Theorem: power series solutions at ordinary points

Suppose $x_0$ is an ordinary point of $y^{\prime \prime }+p{y}^{\prime }+qy=0$. Then there is some neighborhood of $x_0$ on which this equation has a power series solution. Any initial values may be given, and the radius of convergence of the solution is at least as big as the smaller of the radii of convergence of $p$ and $q$.