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