Power series

| Main Article
| Discussion
| Related Articles  [?]
| Bibliography  [?]
| External Links  [?]
| Citable Version  [?]

|
|
|
|
|
|
|
|

This editable Main Article is under development and subject to a disclaimer.

[edit intro]

This article is about Power series. For other uses of the term Power, please see Power (disambiguation).

In mathematics, a power series is an infinite series whose terms involve successive powers of a variable, typically with real or complex coefficients. If the series converges, its value determines a function of the variable involved. Conversely, given a function it may be possible to form a power series from successive derivatives of the function: this Taylor series is then a power series in its own right.

Formally, let z be a variable and an be a sequence of real or complex coefficients. The associated power series is

∑n=0∞anzn..

## Contents

* 1 Radius of convergence
* 1.1 Convergence tests
* 2 Algebra of power series
* 3 Formal power series
* 4 Inversion of power series

## Radius of convergence[edit]

Over the complex numbers the series will have a radius of convergence R, a real number with the property that the series converges for all complex numbers z with |z|<R and that R is the "largest" number with this property (supremum of all numbers with this property. If the series converges for all complex numbers, we formally say that the radius of convergence is infinite.

For example

∑n!zn converges only for z=0 and has radius of convergence zero.
∑zn converges for all |z|<1, but diverges for z=1 and so has radius of convergence 1.
∑zn/n! converges for all complex numbers z and so has radius of convergence infinity.

More generally we may consider power series in a complex variable z−a for a fixed complex number a.

Within the radius of convergence, a power series determines an analytic function of z. Derivatives of all orders exist, and the Taylor series exists and is equal to the original power series.

### Convergence tests[edit]

Some of the standard test for convergence of series translate into computations of the radius of convergence R.

* D'Alembert ratio test: if the limit of the sequence |an+1an| exists, then this is equal to 1/R.
* Cauchy n-th root test: if the limit of the sequence |an|1/n exists, then this is equal to 1/R.

## Algebra of power series[edit]

Power series may be added and multiplied. If ∑anzn and ∑bnzn are power series, we may define their sum and product

(∑anzn)+(∑bnzn)=∑(an+bn)zn
(∑anzn)⋅(∑bnzn)=∑n=0∞(∑k=0nakbn−k)zn.

and these purely algebraic definitions are consistent with the values achieved within the region of convergence.

If a power series g has constant term b0=0, then the n-th power of g involves only powers of z with exponent at least n. Hence if f denotes the series ∑anzn it makes sense to consider the composite

f(g)=∑n=0∞angn

as a power series in z, since any given power of z will appear in only finitely many of the terms gn. Again this purely algebraic definition is consistent with function composition within the region of convergence.

## Formal power series[edit]

Let R be any ring. A formal power series over R, with variable X is a formal sum ∑anXn with coefficients an∈R. Addition and multiplication are now defined purely formally, with no questions of convergence, by the formulae above. The formal power series form another ring denoted R[[X]].

## Inversion of power series[edit]

The power series f is called inverse series of the power series g, iff all elements of the expansion of f(g(z))−z with respect to z are zero.

To simplify formulas, it is assumed that the zero-th element is zero, and the first coefficient is unity: f0=0, and f1=1. Then g0=0, and g1=1, and

g2=−f2
g3=2f22−f3
g4=−5f23+5f3f2−f4
g5=6f4f2+14f24−21f3f2+3f32−f5
g6=7f5f2+84f3f23−28f32f2+7f3f4−28f4f22−42f25−f6
g7=−36f5f2+8f5f3+8f6f2+120f4f23−72f4f3f2+4f42+132f26−330f3f24+180f32f22−12f32−f7

and so on.