Power series

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

$f(x)=\sum _{n=0}^{\infty }a_{n}\left(x-a\right)^{n}$

where the coefficients a_n, the center a, 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.

Radius of convergence

A power series will converge for some values of the variable x (at least for x = a) 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 − a| < r and diverges whenever |x − a| > r. (For |x - a| = r we cannot make any general statement.) The number r is called the radius of convergence of the power series; in general it is given as

$r=\liminf _{n\to \infty }\left|a_{n}\right|^{-1}$

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

$r=\lim _{n\to \infty }\left|{a_{n} \over a_{n+1}}\right|.$

The latter formula is valid only if the limit exists, while the former formula can always be used.

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

Operations on power series

Addition and subtraction

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

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

$g(x)=\sum _{n=0}^{\infty }b_{n}(x-a)^{n}$

then

$f(x)\pm g(x)=\sum _{n=0}^{\infty }(a_{n}\pm b_{n})(x-a)^{n}$

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:

$f(x)g(x)=\left(\sum _{n=0}^{\infty }a_{n}(x-a)^{n}\right)\left(\sum _{n=0}^{\infty }b_{n}(x-a)^{n}\right)$

$=\sum _{i=0}^{\infty }\sum _{j=0}^{\infty }a_{i}b_{j}(x-a)^{n}$

For division, observe:

${f(x) \over g(x)}={\sum _{n=0}^{\infty }a_{n}(x-a)^{n} \over \sum _{n=0}^{\infty }b_{n}(x-a)^{n}}=\sum _{n=0}^{\infty }c_{n}(x-a)^{n}$

$f(x)=\left(\sum _{n=0}^{\infty }b_{n}(x-a)^{n}\right)\left(\sum _{n=0}^{\infty }c_{n}(x-a)^{n}\right)$

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:

$f^{\prime }(x)=\sum _{n=1}^{\infty }a_{n}n\left(x-a\right)^{n-1}$

$\int f(x)\,dx=\sum _{n=0}^{\infty }{\frac {a_{n}\left(x-a\right)^{n+1}}{n+1}}+C$

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

$a_{n}={\frac {f^{\left(n\right)}\left(a\right)}{n!}}$

where f ⁽ⁿ⁾(a) denotes the n-th derivative of f at a. 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 a∈U such that f ⁽ⁿ⁾(a) = g ⁽ⁿ⁾(a) 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 - a| < 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

$f(x_{1},...,x_{n})=\sum _{j_{1},...,j_{n}=0}^{\infty }a_{j_{1},...,j_{n}}\prod _{k=1}^{n}\left(x_{k}-c_{k}\right)^{j_{k}},$

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

$f(x)=\sum _{\alpha \in \mathbb {N} ^{n}}a_{\alpha }\left(x-c\right)^{\alpha }.$

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.