Power series

An infinite series of the form ∑ aₙ(x−c)ⁿ, with coefficients aₙ and center c. It converges for some values of x and diverges for others, usually with a boundary determined by its radius of convergence.

Radius of convergence: The distance from a power series’ center within which the series converges absolutely. It marks the interval or disk where the series defines a function.
Infinite series: The sum of an infinite sequence of terms, defined through limits of its partial sums. A power series is a special kind of infinite series.
Taylor polynomial: A finite polynomial formed by truncating a Taylor series. Truncations provide computable approximations to a power series.
Laurent series: A series expansion that allows both positive and negative integer powers of the variable. Unlike an ordinary power series, it can represent functions near isolated singularities.
Cauchy–Hadamard theorem: A theorem that computes a power series’ radius of convergence from the growth rate of its coefficients. It provides a general method for finding the convergence boundary.
Taylor series: A power series whose coefficients are determined by a function’s derivatives at a chosen point. It is the standard way to obtain a power series from a function.
Numerical analysis: The study of algorithms for obtaining approximate numerical solutions. Truncated power series can approximate functions in numerical calculations.
Fourier series: An expansion in sines and cosines, or equivalent complex exponentials, with integer frequencies. It uses periodic basis functions rather than successive powers of a variable.
Term-by-term differentiation: Differentiation of a series by differentiating each term, valid under suitable convergence conditions. Inside its convergence radius, a power series can be differentiated this way.
Maclaurin series: A Taylor series centered at zero. It is the common special case matching the form ∑ aₙxⁿ.

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