On the coefficients of a typically-real function

Abstract

Now for i sufficiently large all the terms within the last parentheses except the first are as small as we please.Hence for sufficiently large i the difference in question is positive.From this contradiction the theorem follows.In conclusion, we may note as a simple corollary of the above theorem that if lim,^ | a n | 1/n = 1, then lim n -.oo|fln+i/ a n| =1 if and only if there exists a sequence of real numbers \ n such that lim nH>0O X" = l andïïïn^ool |a n +i| -X w |a n | | 1/n <l.