Power series

In mathematics, a power series (in one variable) is an infinite series of the form

<math>

f(x) = \sum_{n=0}^\infty a_n \left( x-c \right)^n = a_0 + a_1 (x-c) + a_2 (x-c)^2 + a_3 (x-c)^3 + \ldots <math>

where the coefficients a_n, the center c, and the argument x are usually real or complex numbers. These series usually arise as the Taylor series of some known function; the Taylor series article contains many examples.

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

<math>

f(x) = \sum_{n=0}^\infty a_n x^n = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \ldots. <math> These power series arise primarily in analysis, but also occur in combinatorics (under the name of generating functions) and in electrical engineering (under the name of the Z-transform). The familiar decimal notation for integers can also be viewed as an example of a power series, but with the argument x fixed at 10. In number theory, the concept of p-adic numbers is also closely related to that of a power series.

Contents

1 Examples 2 Radius of convergence 3 Operations on power series 3.1 Addition and subtraction 3.2 Multiplication and division 3.3 Differentiation and integration 4 Analytic functions 5 Formal power series 6 Power series in several variables 7 Order of a power series

Examples

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 <math>f(x) = x^2 + 2x + 3<math> can be written as a power series around the center <math>c=0<math> as

<math>f(x) = 3 + 2 x + 1 x^2 + 0 x^3 + 0 x^4 + \ldots<math>

or around the center <math>c=1<math> as

<math>f(x) = 6 + 4 (x-1) + 1(x-1)^2 + 0(x-1)^3 + 0(x-1)^4 + \ldots<math>

or indeed around any other center c. One can view power series as being like "polynomials of infinite degree," although power series are not polynomials.

The geometric series formula

<math> \frac{1}{1-x} = \sum_{n=0}^\infty x^n = 1 + x + x^2 + x^3 + \ldots,<math>

which is valid for <math>|x|<1<math>, is one of the most important examples of a power series, as is the exponential function formula

<math> e^x = \sum_{n=0}^\infty \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \ldots.<math>

These power series were also examples of Taylor series. However, there exist power series which are not the Taylor series of any function, for instance

<math>\sum_{n=0}^\infty n! x^n = 1 + x + 2! x^2 + 3! x^3 + \ldots.<math>

Negative powers are not permitted in a power series, for instance <math>1 + x^{-1} + x^{-2} + \ldots<math> is not considered a power series (although it is a Laurent series). Similarly, fractional powers such as <math>x^{1/2}<math> are not permitted (but see Pusieux series). The coefficients <math>a_n<math> are not allowed to depend on <math>x<math>, thus for instance <math>\sin(x) x + \sin(2x) x^2 + \sin(3x) x^3 + \ldots<math> is not a power series.

Radius of convergence

A power series will converge for some values of the variable x (at least for x = c) and may diverge for others. It turns out that 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

<math>r=\liminf_{n\to\infty} \left|a_n\right|^{-\frac{1}{n}}<math>

(see lim inf) but a fast way to compute it is

<math>r=\lim_{n\to\infty}\left|{a_n \over a_{n+1}}\right|.<math>

The latter formula is valid only if the limit exists. The former formula is always correct.

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.

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:

<math>f(x) = \sum_{n=0}^\infty a_n (x-c)^n<math>

<math>g(x) = \sum_{n=0}^\infty b_n (x-c)^n<math>

then

<math>f(x)\pm g(x) = \sum_{n=0}^\infty (a_n \pm b_n) (x-c)^n<math>

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:

<math> f(x)g(x) = \left(\sum_{n=0}^\infty a_n (x-c)^n\right)\left(\sum_{n=0}^\infty b_n (x-c)^n\right)<math>

<math> = \sum_{i=0}^\infty \sum_{j=0}^\infty a_i b_j (x-c)^{i+j}<math>

<math> = \sum_{n=0}^\infty (\sum_{i=0}^n a_i b_{n-i}) (x-c)^n.<math>

The sequence <math>m_n := \sum_{i=0}^n a_i b_{n-i}<math> is known as the convolution of the sequence <math>a_n<math> and <math>b_n<math>.

For division, observe:

<math> {f(x)\over g(x)} = {\sum_{n=0}^\infty a_n (x-c)^n\over\sum_{n=0}^\infty b_n (x-c)^n} = \sum_{n=0}^\infty d_n (x-c)^n<math>

<math> f(x) = \left(\sum_{n=0}^\infty b_n (x-c)^n\right)\left(\sum_{n=0}^\infty d_n (x-c)^n\right)<math>

and then use the above, comparing coefficients.

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:

<math>

f^\prime (x) = \sum_{n=1}^\infty a_n n \left( x-c \right)^{n-1} <math>

<math>

\int f(x)\,dx = \sum_{n=0}^\infty \frac{a_n \left( x-c \right)^{n+1}} {n+1} + C <math>

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

Analytic functions

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

<math>

a_n = \frac {f^{\left( n \right)}\left( c \right)} {n!} <math>

where f ⁽ⁿ⁾(c) denotes the n-th derivative of f at 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 - a| = 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.

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 combinatorics.

Power series in several variables

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

<math>

f(x_1,...,x_n) = \sum_{j_1,...,j_n = 0}^{\infty}a_{j_1,...,j_n} \prod_{k=1}^n \left(x_k - c_k \right)^{j_k}, <math>

where j = (j₁, ..., j_n) is a vector of natural numbers, the coefficients a_(j1,...,jn) are usually real or complex numbers, and the center c = (c₁, ..., c_n) and argument x = (x₁, ..., x_n) are usually real or complex vectors. In the more convenient multi-index notation this can be written

<math>

f(x) = \sum_{\alpha \in \mathbb{N}^n} a_{\alpha} \left(x - c \right)^{\alpha}. <math>

The theory of such series is trickier than for single-variable series. For instance, the region of absolute convergence is now given by a log-convex set rather than an interval. 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.

Order of a power series

Let α be a multi-index for a power series f(x₁, x₂, …, 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.de:Potenzreihe he:טור חזקות nl:Machtreeks ja:冪級数 pl:Szereg potęgowy