proof of Tauber’s convergence theorem



be a complex power series, convergent in the open disk |z|<1. We suppose that

  1. 1.

    nan0 as n, and that

  2. 2.

    f(r) converges to some finite L as r1-;

and wish to show that nan converges to the same L as well.

Let sn=a0++an, where n=0,1,, denote the partial sums of the series in question. The enabling idea in Tauber’s convergence result (as well as other Tauberian theoremsMathworldPlanetmath) is the existence of a correspondence in the evolution of the sn as n, and the evolution of f(r) as r1-. Indeed we shall show that

|sn-f(n-1n)|0asn. (1)

The desired result then follows in an obvious fashion.

For every real 0<r<1 we have




and noting that


we have that


Setting r=1-1/n in the above inequalityMathworldPlanetmath we get




are the Cesàro means of the sequence |kak|,k=0,1, Since the latter sequence converges to zero, so do the means μn, and the suprema ϵn. Finally, Euler’s formula for e gives


The validity of (1) follows immediately. QED

Title proof of Tauber’s convergence theorem
Canonical name ProofOfTaubersConvergenceTheorem
Date of creation 2013-03-22 13:08:20
Last modified on 2013-03-22 13:08:20
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 7
Author rmilson (146)
Entry type Proof
Classification msc 40G10