Series expansion

A representation of a function or quantity as an infinite sum of terms, often involving powers of a variable. Its usefulness depends on where and in what sense the series converges.

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Convergence of a series: The property that the partial sums of an infinite series approach a finite limit. An expansion represents its target only where its series converges to the intended value.
Power series: An infinite sum of constant coefficients multiplied by nonnegative integer powers of a variable. This is the standard form of an expansion in powers of a variable.
Numerical analysis: The study of algorithms for obtaining approximate numerical solutions to mathematical problems. Truncated expansions provide computable approximations with estimable errors.
Continued fraction: A nested expression formed by repeated division, often used to represent numbers or functions. It can represent or approximate functions where a power series is inefficient.
Isaac Newton: English mathematician and physicist who developed foundational methods in calculus and mechanics. Newton used infinite series extensively in early calculus and mathematical physics.
Radius of convergence: The radius of the largest disk around a power series' center where it converges. It identifies the region where a power-series expansion converges.
Taylor series: A power series whose coefficients are determined by a function's derivatives at one point. It constructs a local expansion from derivative data.
Perturbation theory: Methods for approximating solutions by expanding in a small parameter. Its approximations are organized as series in the perturbation strength.
Chebyshev approximation: Polynomial approximation using expansions in Chebyshev polynomials, often optimized over an interval. It often gives better finite-interval approximations than truncating a Taylor series.
Gottfried Wilhelm Leibniz: German mathematician and philosopher who developed calculus notation and methods independently of Newton. Leibniz helped establish the calculus framework in which series expansions became central.