machtseries

Machtseries is a term used in some mathematical texts to denote a power series, a formal series in powers of a variable. The word is not standard in most references; it appears occasionally as a portable label, sometimes chosen for its etymological hint that the series encodes 'power' terms.

Form. A machtseries in variable x is written ∑_{n=0}^∞ c_n x^n, with coefficient sequence (c_n). The radius of convergence R is defined by R = 1 / limsup_{n→∞} |c_n|^{1/n} (with R = ∞ if c_n grows subexponentially or is identically zero). The series converges to an analytic function on the disk |x| < R.

Examples. If c_n = a for all n, machtseries reduces to a ∑ x^n = a/(1-x) for |x|<1. If c_n = 1/n!, the machtseries is e^x. Differentiation or integration termwise yields another machtseries with modified coefficients.

Relation and usage. In combinatorics and mathematical analysis, machtseries function as generating functions encoding sequences. They allow operations such as addition, multiplication, and composition, under appropriate convergence. The term is largely synonymous with the standard term 'power series' and is used infrequently.

See also. Power series; Generating function; Analytic function.