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Jose27 · #3 July 3rd, 2009, 05:01 PM

Quote:

Originally Posted by wellfed

Hi I am stuck on the following problem, your help is greatly appreciated.

given $a_n > 0$ and $lim(n a_n) = L$ with $L \ne 0$ show the series $\Sigma a_n$ diverges.

Suppose $0 \leq a_1 \leq ...\leq a_n \leq ...$ and that $\lim_{n \rightarrow \infty} na_n$ $=L$ then $\sum_{i=1}^n {a_i}$ $\leq na_n \leq L$ for all $n \in \mathbb{N}$ and so the series converges.

but then a_n=0 for all n \in \mathbb{N} so it doesn't work. I'll check it later.