proof of Abel’s convergence theorem

Suppose that


is a convergent seriesMathworldPlanetmathPlanetmath, and set


Convergence of the first series implies that an0, and hence f(r) converges for |r|<1. We will show that f(r)L as r1-.



denote the corresponding partial sums. Our proof relies on the following identity

f(r)=nanrn=(1-r)nsnrn. (1)

The above identity obviously works at the level of formal power series. Indeed,


Since the partial sums sn converge to L, they are bounded, and hence nsnrn converges for |r|<1. Hence for |r|<1, identity (1) is also a genuine functional equality.

Let ϵ>0 be given. Choose an N sufficiently large so that all partial sums, sn with n>N, satisfy |sn-L|ϵ. Then, for all r such that 0<r<1, one obtains


Note that


As r1-, the first term tends to 0. The absolute valueMathworldPlanetmathPlanetmathPlanetmath of the second term is estimated by ϵrN+1ϵ. Hence,

lim supr1-|f(r)-L|ϵ.

Since ϵ>0 was arbitrary, it follows that f(r)L as r1-. QED

Title proof of Abel’s convergence theorem
Canonical name ProofOfAbelsConvergenceTheorem
Date of creation 2013-03-22 13:07:39
Last modified on 2013-03-22 13:07:39
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 9
Author rmilson (146)
Entry type Proof
Classification msc 40G10
Related topic ProofOfAbelsLimitTheorem