
Fractions ad infinitum
Posted:
Jun 26, 2013 3:26 AM


Let a be a sequence into (0,oo) (a1 in R permitted) and define by induction [a1] = a1; [a1, a2] = a1 + 1/a2 [a1,.. a_(j+1)] = [a1,.. a_(j1), aj + 1/a_(j+1)] [a1,a2,.. ] = lim(j>oo) [a1,.. aj]
It's know that if sum(j=1,oo) aj = oo, then the continued fraction [a1,a2,.. ] converges. Has anyone any suggestions how to prove the converse, that if [a1,a2,.. ] converges, then sum(j=1,oo) aj diverges?

