Raabe's TheoremDate: 8/23/96 at 4:49:25 From: Manuel Ojeda Aciego Subject: about Raabe's theorem Dear Dr. Math: This question is about Raabe's result for the convergence of a numerical series of non-negative terms. It states the convergence or divergence of the series by means of the result m of the following limit a(n+1) lim n ( 1 - -------- ) = m. n->Infty a(n) If m > 1 then the series converge, if m < 1 then the series diverge, and (here is my problem) if m = 1 nothing can be asserted. The harmonic series is an example of divergent series having m=1, what I am looking for is a convergent series having m = 1, in order to convince myself of the last part of the theorem. Best regards, Manuel Ojeda Aciego Dept. Matematica Aplicada Universidad de Malaga Date: 8/23/96 at 19:39:50 From: Doctor Tom Subject: Re: about Raabe's theorem How about a(n) = 1/(n*log(n)*log(n))? This converges (See Knopp, Infinite Sequences and Series, for example), and with a little work, I think you can convince yourself that the limit of n(1 - a(n+1)/a(n)) is also 1. -Doctor Tom, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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