proof of identity theorem of power series

We can prove the identity theorem for power seriesMathworldPlanetmath using divided differencesDlmfMathworldPlanetmath. From amongst the points at which the two series are equal, pick a sequence {wk}k=0 which satisfies the following three conditions:

  1. 1.


  2. 2.

    wm=wn if and only if m=n.

  3. 3.

    wkz0 for all k.

Let f be the functionMathworldPlanetmath determined by one power series and let g be the function determined by the other power series:

f(z) =n=0an(z-z0)n
g(z) =n=0bn(z-z0)n

Because formation of divided differences involves finite sums and dividing by differences of wk’s (which all differ from zero by condition 2 above, so it is legitimate to divide by them), we may carry out the formation of finite diffferences on a term-by-term basis. Using the result about divided differences of powers, we have

Δmf[wk,,wk+m] =n=manDmnk
Δmf[wk,,wk+m] =n=mbnDmnk



Note that limkinftyDmnk=0 when m>n, but Dmmk=1. Since power series converge uniformly, we may intechange limit and summation to conclude

limkΔmf[wk,,wk+m] =n=manlimkDmnk=am
limkΔmg[wk,,wk+m] =n=mbnlimkDmnk=bm.

Since, by design, f(wk)=g(wk), we have


hence am=bm for all m.

Title proof of identity theorem of power seriesPlanetmathPlanetmath
Canonical name ProofOfIdentityTheoremOfPowerSeries1
Date of creation 2013-03-22 16:48:46
Last modified on 2013-03-22 16:48:46
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 11
Author rspuzio (6075)
Entry type Proof
Classification msc 40A30
Classification msc 30B10