machtserie

Machtserie, known in English as a power series, is an infinite series that represents a function around a chosen center. It has the form sum_{n=0}^∞ a_n (z - c)^n, where c is the center and a_n are coefficients. When the variable is real, z can be replaced by x; when complex, z lies in the complex plane. Power series are fundamental in analysis and are often written as Taylor series when the coefficients are derived from derivatives at the center.

The convergence of a machtserie is governed by a radius of convergence R ≥ 0. The series converges absolutely for |z - c| < R and diverges for |z - c| > R. The radius is given by the Cauchy–Hadamard formula, R = 1 / limsup_{n→∞} |a_n|^{1/n}. On the boundary |z - c| = R, convergence may occur or fail depending on the specific coefficients. Within the disk of convergence, the series defines a holomorphic (complex-analytic) function and, for real variables, a smooth function.

Differentiation and integration are allowed term-by-term inside the disk: f'(z) = sum_{n=1}^∞ n a_n (z - c)^{n-1}, and similar for the integral. A power series is the Taylor series of a function around c when a_n = f^{(n)}(c)/n%; hence the Maclaurin series is the special case centered at 0.

Examples include the geometric series sum_{n=0}^∞ r^n for |r| < 1 and fundamental expansions such as e^x = sum_{n=0}^∞ x^n / n!, sin x, and cos x. In addition to analytic applications, formal power series treat the series algebraically without regard to convergence and are central in combinatorics and algebra.