# Thread: 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

$\displaystyle Problem$ $\displaystyle 1$

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

show that:

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

2. Nice one!

Spoiler:

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

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

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

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

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

3. Elegant and powerful !!!