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MATH
[1251]

Definitions from Power series

All definitions from Power series.

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Power series

A power series centred at aa is a series of the form

∑k=0∞ck(x−a)k.\sum_{k=0}^{\infty}c_k(x-a)^k.

The series itself may define a new function at every value of xx where it converges.

  1. [1]:Radius and interval of convergence
  2. [2]:Algebra of power series
  3. [3]:Differentiate and integrate term by term

Radius and interval of convergence

The radius of convergence RR of

∑k=0∞ck(x−a)k\sum_{k=0}^{\infty}c_k(x-a)^k

is the number such that the series converges absolutely when ∣x−a∣<R|x-a|<R and diverges when ∣x−a∣>R|x-a|>R.

The interval of convergence is the open interval (a−R,a+R)(a-R,a+R) together with whichever endpoints converge.

With R=0R=0 the series converges only at its centre, while R=∞R=\infty gives convergence for every real xx. Here RR denotes a radius; the Taylor remainder is written separately as Rn+1(x)R_{n+1}(x).

  1. [1]:Power series
  2. [2]:Convergence has a central interval

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Power seriesRadius and interval of convergence