proof of Tauber’s convergence theorem
as , and that
converges to some finite as ;
and wish to show that converges to the same as well.
Let , where , denote the partial sums of the series in question. The enabling idea in Tauber’s convergence result (as well as other Tauberian theorems) is the existence of a correspondence in the evolution of the as , and the evolution of as . Indeed we shall show that
The desired result then follows in an obvious fashion.
For every real we have
and noting that
we have that
Setting in the above inequality we get
are the Cesàro means of the sequence Since the latter sequence converges to zero, so do the means , and the suprema . Finally, Euler’s formula for gives
The validity of (1) follows immediately. QED
|Title||proof of Tauber’s convergence theorem|
|Date of creation||2013-03-22 13:08:20|
|Last modified on||2013-03-22 13:08:20|
|Last modified by||rmilson (146)|