Power Series · aiwiki.page

A power series is an infinite series of the form

\sum_{n=0}^{\infty}a_n(x-c)^n,

where $a_n$ are coefficients and $c$ is the center of expansion. The variable and coefficients usually belong to the real numbers or complex numbers. A power series generalizes a polynomial by allowing infinitely many terms. In mathematical analysis, its convergence determines where it defines a function; in algebra, it can instead be treated as a formal expression without assigning numerical values to the variable. (math.ucdavis.edu)

Convergence and radius of convergence

For a fixed value of $x$ , convergence means that the partial sums

S_N(x)=\sum_{n=0}^{N}a_n(x-c)^n

approach a finite limit as $N$ increases. Every power series has a radius of convergence $R$ , with $0\le R\le\infty$ . It converges absolutely when $|x-c|<R$ and diverges when $|x-c|>R$ . If $R=0$ , it converges only at its center; if $R=\infty$ , it converges everywhere. For a complex variable, the interior convergence region is a disk; for a real variable, it is an interval. Boundary points require separate examination. (math.ucdavis.edu)

The Cauchy–Hadamard formula expresses the radius directly in terms of the coefficients:

\frac1R=\limsup_{n\to\infty}|a_n|^{1/n},

with $1/0=\infty$ and $1/\infty=0$ . When the coefficients are eventually nonzero and the indicated limit exists, an alternative formula is

R=\lim_{n\to\infty}\left|\frac{a_n}{a_{n+1}}\right|.

These formulas describe interior convergence but do not settle convergence at the boundary. (math.ucdavis.edu)

Representative examples

The geometric series

1+x+x^2+\cdots=\frac1{1-x},\qquad |x|<1,

has radius $1$ and diverges at both real endpoints. By contrast, the logarithmic expansion

\log(1+x)=\sum_{n=1}^{\infty} \frac{(-1)^{n-1}x^n}{n}

has the same radius but converges at $x=1$ , where its sum is $\log 2$ , and diverges at $x=-1$ . Thus, equal radii do not imply equal endpoint behavior. (math.ucdavis.edu)

The exponential function has the expansion

e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!},

where $n!$ denotes a factorial. This series converges for every real or complex $x$ . Another important example is the generalized binomial expansion:

(1+x)^\alpha =\sum_{n=0}^{\infty}\binom{\alpha}{n}x^n, \qquad |x|<1.

For a nonnegative integer $\alpha$ , the expansion terminates and becomes a polynomial identity valid without this restriction. (dlmf.nist.gov)

Differentiation and integration

Inside its radius of convergence, a power series has uniform convergence on every closed interval, or closed complex disk, lying strictly inside the convergence region. Its sum is consequently a continuous function. More strongly, it can be differentiated and integrated term by term:

f'(x)=\sum_{n=1}^{\infty}n a_n(x-c)^{n-1},

\int_c^x f(t)\,dt =\sum_{n=0}^{\infty}\frac{a_n}{n+1}(x-c)^{n+1}.

The derivative series and integral series retain the original radius, although their boundary behavior can differ. Repeated differentiation is also valid throughout the interior. These properties allow many operations familiar from finite polynomials to extend to infinite expansions. (math.ucdavis.edu)

Taylor series and analyticity

If a power series has positive radius and sums to $f$ , its coefficients are uniquely determined:

a_n=\frac{f^{(n)}(c)}{n!}.

It is therefore the Taylor series of its own sum. A Taylor series centered at zero is called a Maclaurin series. An analytic function is one that equals a convergent power series in a neighborhood of each point of its domain. In complex analysis, every holomorphic function is analytic: if it is holomorphic throughout $|z-c|<r$ , its Taylor expansion there has radius at least $r$ . (dlmf.nist.gov)

For real functions, possessing derivatives of every order is not sufficient for analyticity. For example,

f(x)= \begin{cases} e^{-1/x^2},&x\ne0,\\ 0,&x=0 \end{cases}

is infinitely differentiable, but all its derivatives at zero vanish. Its Taylor series at zero is therefore identically zero and does not represent the function nearby. Equality with a Taylor series requires the approximation remainder to tend to zero, not merely the existence of its coefficients. (math.ucdavis.edu)

Algebraic and formal operations

Two convergent power series with the same center can be added coefficient by coefficient. Their product is given by the Cauchy product: if their coefficients are $a_n$ and $b_n$ , the product coefficient is

d_n=\sum_{k=0}^{n}a_kb_{n-k}.

Both operations converge at least inside the smaller original radius; cancellation may produce a larger radius. A series whose constant coefficient is nonzero also has a convergent reciprocal expansion sufficiently near its center. (math.ucdavis.edu)

A formal power series retains these coefficient rules but does not require numerical convergence. With coefficients in a commutative ring $A$ , such expressions form the ring $A[[x-ray-crystallography|x]]$ . Each product coefficient involves only finitely many terms, so multiplication remains well defined even for series that diverge at every nonzero numerical argument. (ocw.mit.edu)

Applications

Power series provide local representations and polynomial approximations of functions. In solving a differential equation, substituting an unknown series and equating coefficients often yields a recurrence relation that determines successive terms. Establishing convergence then connects the coefficient calculation to an actual solution. (ocw.mit.edu)

In enumerative mathematics, a generating function encodes a sequence as coefficients. For example, if $a_0=a_1=1$ and $a_n=a_{n-1}+a_{n-2}$ , then

\sum_{n=0}^{\infty}a_nx^n=\frac1{1-x-x^2}

as a formal identity. Here the coefficients are shifted Fibonacci numbers, and the rational expression packages their recurrence into an algebraic equation. (math.mit.edu)