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March 2nd, 2009, 12:10 PM   #1
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Confused

I am given a_n > 0, lim (n*a_n) = l, l != 0... show that \Sum (a_n) is divergent.

I would like a hint where to start. I thought that since the limit n*a_n = l, that means that n*a_n is bounded by some value greater than 0... but I am unsure if that can help me show that a_n is unbounded. Or am I completely in the wrong area here.

Thank you
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March 2nd, 2009, 01:35 PM   #2
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Re: Confused

If and then so by the comparison of limits test, converges iff converges.
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March 2nd, 2009, 03:52 PM   #3
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Re: Confused

Thanks... The limit comparison test isn't a theorem we can use, but it gave me an idea on how to do the problem

if a_n >0 for all and and lim (n*a_n) = l, that implies that n*a_n is bounded. Thus there exist real numbers c,d such that
c <= n*a_n <= d <=> 1/n * c <= a_n. Since Sum (1/n) diverges (sum c(1/n) diverges) then by the comparison test Sum(a_n) must diverge.

by similar logic if lim n^2*a_n, converges, a_n > 0 then Sum (a_n) must converge.
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