dodgy 'proof'

hello,

My 'proof' for the following is pretty bad (if it works at all), so i'm
looking for clarification:

the problem is to show that for a[n]>0, if sum a[n] converges then lim inf
(n*a[n]) = 0. where lim inf - is meant to be the least of the limits.

I know that if lim sup |a[n]|^1/n <1 then the series is absolutely
convergent. Since a[n]>0 and is here assumed convergent, then we
automaticaly have this fact. For let x[n] =
sup{a[n]^1/n,a[n+1]^1/(n+1),...} then if lim sup a[n]^1/n =p<1 then taking
e=(1-p)/2 there exists N such that n>=N where p-e<x[n]<p+e. So x[n]
<(p+1)/2 <1. Threfore for all n>=N a[n]^1/n <= x[n] <1 => n*a[n] <=
n*x[n]^n. Since x[n]<1, this goes to zero in the limit. But the limit in
the qustion is the lim inf - why?



.

Relevant Pages

  • Re: dodgy proof
    ... looking for clarification: ... convergent. ... David C. Ullrich ...
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    ... looking for clarification: ... limits. ... However robert said "There are convergent ...
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    ... looking for clarification: ... and the series is therefore not convergent, which is a contradiction. ...
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