# Math Help - Summon the Heroes-1

1. ## 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

$Problem$ $1$

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

show that:

$\lim_{n \rightarrow \infty}\frac{a_{n}}{(1+a_{1})(1+a_{2})...(1+a_{n}) }=0$

2. 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

$\frac{a_n}{(1+a_1)\cdots(1+a_n)}\leq \frac{1}{(1+a_1)\cdots(1+a_{n-1})}\to 0$

3. Elegant and powerful !!!