Math 2.0

definition

Power series

Definition

Let cc be a real number and let (an)n≥0(a_n)_{n \ge 0} be a real sequence. The power series with center cc and coefficients ana_n is the series ∑n=0∞an(x−c)n,\sum_{n=0}^{\infty} a_n (x - c)^n, taken as a function of the real variable xx.

Remarks

The series always converges at x=cx = c, with sum a0a_0. The set of xx where it converges is an interval around cc. Its half length is the radius of convergence. Polynomials are power series with only finitely many nonzero coefficients. Complex power series use the same definition, with a complex variable zz.

Explanations

L1 human

A power series is a polynomial with infinitely many terms. For each xx, we put xx into every term and add up the resulting series of numbers.

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Sequence, definition, no evidence levelSequencedefinitionReal sequence, definition, no evidence levelReal sequencedefinitionSeries, definition, no evidence levelSeriesdefinitionPower series, definition, no evidence levelPower seriesdefinition
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Sources

  1. Web https://en.wikipedia.org/wiki/Power_series, lead section

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