Thread: Equivalent

Hi everybody,

Any idea to solve this problem: One denotes $\displaystyle \mathbb N=\{0,1,\cdots\}$.

Let $\displaystyle (L_n)_{n\in\mathbb N}$ be a positive non-decreasing sequence such that $\displaystyle L_n\underset{n\to+\infty}{\sim}q^n$ with $\displaystyle q>0$.

Then, for every $\displaystyle n\in\mathbb{N}$
$\displaystyle \sum_{\substack{u_1+\cdots+u_{n+1}=j-n\\u_1\ge0,\cdots,u_{n+1}\ge0}}\limits L_{u_1}L_{u_2}\cdots L_{u_{n+1}}\underset{j\to+\infty}{\sim}L_j\binom{j }{n}.$

Thanks in advance