On the Generalized Newtonian Binomial Theorem

Abstract

In this paper, a generalized binomial theorem about the real function (1+t)^α (α≠ 0, 1, 2, . . . ) is proposed, which is proved to be convergent to (1 + t)^α in the region -1<t<-2/â„-1(â„>0) for all real values of α(α≠0,1,2, . . .), and even in the region -2/â„-1<t<-1(â„>0) for such values of α that (1+t)^α has meanings for t<-1, so that it can be convergent to (1+t)^α in the whole region where (1+t)^α has meanings. Moreover, the classical Newtonian binomial expression is a special case of it at â„=-1.