Power series

A power series encoding a sequence through its coefficients. It turns combinatorial splitting and concatenation rules into algebraic identities. A counting generating function is not necessarily a probability generating function: its coefficients need not sum to one.

An exponential generating function encodes a sequence using the displayed factorial denominators. It is especially useful for counting labelled combinatorial structures: taking an unordered set of structures with positive size corresponds to exponentiating their exponential generating function. The rooted-tree generating function illustrates this through $T=z{e}^T$.

A power series is lacunary when only a sparse sequence of powers has nonzero coefficients. For ${\sum }_j{c}_j{z}^{m_j}$ with strictly increasing integer exponents, the radius of convergence satisfies $R^{-1}={\mathrm{limsup}}_j|c_j{|}^{1/m_j}$ by the Cauchy-Hadamard theorem. The exponent $m_j$, rather than the term index $j$, controls the coefficient root.

For $|x|<1$, the binomial series is

For a nonnegative integer $L$ and $|z|<1$, this specialization of the binomial series sums the normalizing coefficients of a negative binomial distribution.

A power series $g(x)={\sum }_{\alpha }{b}_{\alpha }{x}^{\alpha }$ with nonnegative coefficients majorizes $f(x)={\sum }_{\alpha }{a}_{\alpha }{x}^{\alpha }$ when $|a_{\alpha }|\le b_{\alpha }$ for every multi-index $\alpha$. Coefficientwise majorization turns estimates for an analytic equation into estimates for a simpler positive-coefficient equation.

For a series ${\sum }_n{a}_n{x}^{m_n}$ with distinct increasing integer exponents, define the ordinary coefficients to be $a_n$ at $m_n$ and zero elsewhere. The Cauchy-Hadamard theorem then givesThe exponent in the root is the actual degree $m_n$, not the index $n$. For $a_n=(n!{)}^2$ and $m_n=n^2$, $0\le 2\log (n!)/n^2\le 2\log n/n\to 0$, so $R=1$. Boundary behavior is a separate question.

For a nonzero polynomial $p$, the power series ${\sum }_{n\ge 0}p(n)z^n$ has radius of convergence $1$. If $p$ has degree $d$ and leading coefficient $c\ne 0$, then $p(n)\sim c{n}^d$ and $|p(n+1)/p(n)|\to 1$, so the ratio test applies inside the unit disk. At every point of its boundary, the terms fail to tend to zero. The zero polynomial is the exceptional case with radius of convergence $\infty$.

Suppose real coefficients satisfy $c\le a_n\le C(1+n{)}^m$ for all sufficiently large $n$, where $c,C>0$ and $m\ge 0$. Then the power series $\sum a_n{z}^n$ has radius of convergence one. For $|z|<1$, the comparison test for series and the ratio test dominate it by a polynomial times a geometric sequence. For $|z|\ge 1$, the terms have modulus at least $c$ eventually and violate the term test for divergence. Finite changes to the coefficients do not affect the conclusion.

For $\sum a_n{z}^n$,with the usual conventions for zero and infinity.

If $L=\mathrm{limsup}{a}_n^{1/n}$ for positive coefficients, then the coefficients $a_n^2$ have root limsup $L^2$. Coefficients $a_n^{a_n}$ instead have root termsso their radius depends on the growth of $a_n\log a_n$ relative to $n$.

If the ordinary power series ${\sum }_{n\ge 0}{b}_n{w}^n$ has radius $R\in (0,\infty )$, thenconverges for $\mathrm{Re}z<\log R$ and diverges for $\mathrm{Re}z>\log R$. Behavior on the boundary depends on the coefficients.

The Cauchy product has coefficients $c_n={\sum }_{k=0}^n{a}_k{b}_{n-k}$. Absolute convergence permits regrouping and makes its sum the product of the two sums.

Inside its radius of convergence, a power series may be differentiated term by term, and the differentiated series has the same radius.

The first nonzero Taylor term controls a holomorphic function’s local magnitude and angular sign pattern.