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In mathematics, a power series (in one variable) is an infinite series of the form_{n} represents the coefficient of the nth term, c is a constant, and x varies around c (for this reason one sometimes speaks of the series as being centered at c). This series usually arises as the Taylor series of some known function; the Taylor series article contains many examples.## Examples

## Radius of convergence

## Operations on power series

### Addition and subtraction

When two functions f and g are decomposed into power series around the same center c, the power series of the sum or difference of the functions can be obtained by termwise addition and subtraction. That is, if:
### Multiplication and division

With the same definitions above, for the power series of the product and quotient of the functions can be obtained as follows:### Differentiation and integration

Once a function is given as a power series, it is continuous wherever it converges and is differentiable on the interior of this set. It can be differentiated and integrated quite easily, by treating every term separately:## Analytic functions

## Formal power series

In abstract algebra, one attempts to capture the essence of power series without being restricted to the fields of real and complex numbers, and without the need to talk about convergence. This leads to the concept of formal power series, a concept of great utility in algebraic combinatorics.
## Power series in several variables

## Order of a power series

Let α be a multi-index for a power series f(x_{1}, x_{2}, …, x_{n}). The order of the power series f is defined to be the least value |α| such that a_{α} ≠ 0, or 0 if f ≡ 0. In particular, for a power series f(x) in a single variable x, the order of f is the smallest power of x with a nonzero coefficient. This definition readily extends to Laurent series.
## External links

- $f(x)\; =\; sum\_\{n=0\}^infty\; a\_n\; left(x-c\; right)^n\; =\; a\_0\; +\; a\_1\; (x-c)^1\; +\; a\_2\; (x-c)^2\; +\; a\_3\; (x-c)^3\; +\; cdots$

In many situations c is equal to zero, for instance when considering a Maclaurin series. In such cases, the power series takes the simpler form

- $$

Any polynomial can be easily expressed as a power series around any center c, albeit one with most coefficients equal to zero. For instance, the polynomial $f(x)\; =\; x^2\; +\; 2x\; +\; 3$ can be written as a power series around the center $c=0$ as

- $f(x)\; =\; 3\; +\; 2\; x\; +\; 1\; x^2\; +\; 0\; x^3\; +\; 0\; x^4\; +\; cdots\; ,$

- $f(x)\; =\; 6\; +\; 4\; (x-1)\; +\; 1(x-1)^2\; +\; 0(x-1)^3\; +\; 0(x-1)^4\; +\; cdots\; ,$

The geometric series formula

- $frac\{1\}\{1-x\}\; =\; sum\_\{n=0\}^infty\; x^n\; =\; 1\; +\; x\; +\; x^2\; +\; x^3\; +\; cdots,$

- $e^x\; =\; sum\_\{n=0\}^infty\; frac\{x^n\}\{n!\}\; =\; 1\; +\; x\; +\; frac\{x^2\}\{2!\}\; +\; frac\{x^3\}\{3!\}\; +\; cdots,$

- $sin(x)\; =\; sum\_\{n=0\}^infty\; frac\{(-1)^n\; x^\{2n+1\}\}\{(2n+1)!\}\; =\; x\; -\; frac\{x^3\}\{3!\}\; +\; frac\{x^5\}\{5!\}\; -\; frac\{x^7\}\{7!\}+cdots,$

Negative powers are not permitted in a power series, for instance $1\; +\; x^\{-1\}\; +\; x^\{-2\}\; +\; cdots$ is not considered a power series (although it is a Laurent series). Similarly, fractional powers such as $x^\{1/2\}$ are not permitted (but see Puiseux series). The coefficients $a\_n$ are not allowed to depend on $x$, thus for instance:

- $sin(x)\; x\; +\; sin(2x)\; x^2\; +\; sin(3x)\; x^3\; +\; cdots\; ,$ is not a power series.

A power series will converge for some values of the variable x and may diverge for others. All power series will converge at x = c. There is always a number r with 0 ≤ r ≤ ∞ such that the series converges whenever |x − c| < r and diverges whenever |x − c| > r. The number r is called the radius of convergence of the power series; in general it is given as

- $r=liminf\_\{ntoinfty\}\; left|a\_nright|^\{-frac\{1\}\{n\}\}$

$r^\{-1\}=limsup\_\{ntoinfty\}\; left|a\_nright|^\{frac\{1\}\{n\}\}$ (see limit superior and limit inferior). A fast way to compute it is

- $r^\{-1\}=lim\_\{ntoinfty\}left|\{a\_\{n+1\}over\; a\_n\}right|$

if this limit exists.

The series converges absolutely for |x - c| < r and converges uniformly on every compact subset of {x : |x − c| < r}.

For |x - c| = r, we cannot make any general statement on whether the series converges or diverges. However, Abel's theorem states that the sum of the series is continuous at x if the series converges at x.

- $f(x)\; =\; sum\_\{n=0\}^infty\; a\_n\; (x-c)^n$

- $g(x)\; =\; sum\_\{n=0\}^infty\; b\_n\; (x-c)^n$

- $f(x)pm\; g(x)\; =\; sum\_\{n=0\}^infty\; (a\_n\; pm\; b\_n)\; (x-c)^n.$

- $f(x)g(x)\; =\; left(sum\_\{n=0\}^infty\; a\_n\; (x-c)^nright)left(sum\_\{n=0\}^infty\; b\_n\; (x-c)^nright)$

- $=\; sum\_\{i=0\}^infty\; sum\_\{j=0\}^infty\; a\_i\; b\_j\; (x-c)^\{i+j\}$

- $=\; sum\_\{n=0\}^infty\; left(sum\_\{i=0\}^n\; a\_i\; b\_\{n-i\}right)\; (x-c)^n.$

The sequence $m\_n\; =\; sum\_\{i=0\}^n\; a\_i\; b\_\{n-i\}$ is known as the convolution of the sequences $a\_n$ and $b\_n$.

For division, observe:

- $\{f(x)over\; g(x)\}\; =\; \{sum\_\{n=0\}^infty\; a\_n\; (x-c)^noversum\_\{n=0\}^infty\; b\_n\; (x-c)^n\}\; =\; sum\_\{n=0\}^infty\; d\_n\; (x-c)^n$

- $f(x)\; =\; left(sum\_\{n=0\}^infty\; b\_n\; (x-c)^nright)left(sum\_\{n=0\}^infty\; d\_n\; (x-c)^nright)$

and then use the above, comparing coefficients.

- $$

- $$

Both of these series have the same radius of convergence as the original one.

A function f defined on some open subset U of R or C is called analytic if it is locally given by power series. This means that every a ∈ U has an open neighborhood V ⊆ U, such that there exists a power series with center a which converges to f(x) for every x ∈ V.

Every power series with a positive radius of convergence is analytic on the interior of its region of convergence. All holomorphic functions are complex-analytic. Sums and products of analytic functions are analytic, as are quotients as long as the denominator is non-zero.

If a function is analytic, then it is infinitely often differentiable, but in the real case the converse is not generally true. For an analytic function, the coefficients a_{n} can be computed as

- $$

where $f^\{(n)\}(c)$ denotes the nth derivative of f at c, and $f^\{(0)\}(c)\; =\; f(c)$. This means that every analytic function is locally represented by its Taylor series.

The global form of an analytic function is completely determined by its local behavior in the following sense: if f and g are two analytic functions defined on the same connected open set U, and if there exists an element c∈U such that f^{ (n)}(c) = g^{ (n)}(c) for all n ≥ 0, then f(x) = g(x) for all x ∈ U.

If a power series with radius of convergence r is given, one can consider analytic continuations of the series, i.e. analytic functions f which are defined on larger sets than { x : |x - c| < r } and agree with the given power series on this set. The number r is maximal in the following sense: there always exists a complex number x with |x - c| = r such that no analytic continuation of the series can be defined at x.

The power series expansion of the inverse function of an analytic function can be determined using the Lagrange inversion theorem.

An extension of the theory is necessary for the purposes of multivariable calculus. A power series is here defined to be an infinite series of the form

- $$

where j = (j_{1}, ..., j_{n}) is a vector of natural numbers, the coefficients
a_{(j}1,...,j_{n}) are usually real or complex numbers, and the center c = (c_{1}, ..., c_{n}) and argument x = (x_{1}, ..., x_{n}) are usually real or complex vectors. In the more convenient multi-index notation this can be written

- $$

The theory of such series is trickier than for single-variable series, with more complicated regions of convergence. For instance, the power series $sum\_\{n=0\}^infty\; x\_1^n\; x\_2^n$ is absolutely convergent in the set $\{\; (x\_1,x\_2):\; |x\_1\; x\_2|\; <\; 1\}$ between two hyperbolae. (This is an example of a log-convex set, in the sense that the set of points $(log\; |x\_1|,\; log\; |x\_2|)$, where $(x\_1,x\_2)$ lies in the above region, is a convex set. More generally, one can show that when c=0, the interior of the region of absolute convergence is always a log-convex set in this sense.) On the other hand, in the interior of this region of convergence one may differentiate and integrate under the series sign, just as one may with ordinary power series.

- Complex Power Series Module by John H. Mathews
- Powers of Complex Numbers by Michael Schreiber, The Wolfram Demonstrations Project.

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Last updated on Thursday September 25, 2008 at 09:01:22 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Thursday September 25, 2008 at 09:01:22 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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