Summon the Heroes-1

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November 28th 2009, 04:49 AM
Xingyuan

Summon the Heroes-1

From now on, I will post some problem to play with,If this problem be worked out, The next will be posted.

Let's Enjoy the great game...

Have Fun(Rock)

$Problem$ $1$

If $a_{n}>0$ for $\forall$ $n \in \mathbb{N}^+$

show that:

$\lim_{n \rightarrow \infty}\frac{a_{n}}{(1+a_{1})(1+a_{2})...(1+a_{n}) }=0$
November 28th 2009, 05:20 AM
Laurent

Nice one!

Spoiler:

The sequence of general term $(1+a_1)\cdots (1+a_n)$ , $n\geq 1$ , is increasing, therefore either it has a finite limit, or it diverges to $+\infty$ .

If it has a finite limit $\ell$ (note that $1\leq\ell$ ), then $a_n\to 0$ because the ratio $1+a_n$ of two consecutive terms in the previous sequence tends to $\frac{\ell}{\ell}=1$ . Therefore,

$\frac{a_n}{(1+a_1)\cdots(1+a_n)}\leq a_n\to 0.$

And if it diverges to $+\infty$ , then, using $\frac{a_n}{1+a_n}\leq 1$ , we have

$\frac{a_n}{(1+a_1)\cdots(1+a_n)}\leq \frac{1}{(1+a_1)\cdots(1+a_{n-1})}\to 0$
November 28th 2009, 08:25 AM
Xingyuan

Elegant and powerful !!!
(Cool)

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