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Hoi, i wish to show a few things...

Suppose we know $\lim_{n\to\infty}(a_1a_2\cdots a_{n})^{1/n} = \prod_{k=1}^{\infty}\left(1+\frac{1}{k(k+2)}\right)^{\frac{\log k}{\log 2}}$

I hope to show this implies $\frac{a_1+\cdots + a_n}{n}\to\infty $

This is like showing that $a_1+\cdots + a_n$ grows harder then $n^{1+\epsilon}$. Can we conclude for example for large n something like $a_n \approx (1+\frac{1}{n(n+2)})^{n\log(n)/\log(2)}$.

Can someone maybe suggest some ideas?

thank you

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    Are the $a_n$ terms positive integers?2012-11-06
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    yes, all positive integers :)2012-11-06
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    this is the context of the $a_n$ http://en.wikipedia.org/wiki/Khinchin's_constant2012-11-06
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    I think i like to rephrase my question: Can we conclude $\lim_{n\to\infty}\frac{a_1+\cdots + a_n}{n} = \infty$ from the assumption that for some constant $K$ we have $\lim_{n\to\infty} \frac{\log(a_1)+\cdots \log(a_n)}{n}= K$2012-11-06
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    Why couldn't all $a_i$ be equal to (roughly) $e^K$? Then the limit certainly does not diverge.2012-11-07

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No such conclusion is possible: let $K$ be the infinite product $\prod_{k=1}^\infty \big(1+\tfrac{1}{k(k+2)}\big)^{\log k/\log 2}$ (this converges because $\sum (\log k)/k^2$ converges). Then one can choose a sequence of integers $\{a_n\}$ with $a_n$ is very close to $K$ (i.e. $| a_n - K|<1$) such that $(a_1 \cdots a_n)^{1/n} \to K$. On the other hand, $(a_1 + \cdots + a_n)/n$ will always be less than $K+1$.

For problems of this type, something like Jensen's inequality seems like a useful tool to have in one's arsenal.