J.E. Littlewood: 'The Converse of Abel's Theorem on Power Series'

摘要

In 1897, the Austrian mathematician A. Tauber published the short paper (5 pages) [2] which can be summarized as follows:(1) If a series Σa_n converges, i.e., if S_n:= Σ_n~N=0 a_n converges, then lim_(x→1~-)Σ_(n=0)~∞a_nx~n=:lim_(x→1~-)f(x)=Σ_(n=0)~∞a_n by a theorem of Abel.