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Topic: Fractions ad infinitum
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William Elliot

Posts: 2,637
Registered: 1/8/12
Fractions ad infinitum
Posted: Jun 26, 2013 3:26 AM
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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_(j-1), 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?

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