Limit of the series $\lim_{n\rightarrow \infty}\frac{1}{s_n}\sum_{k=1}^{n}a_kx_k$

Question

Let $\{a_n\}$ be a be a strictly increasing sequence of positive real number with limit $\lim_{n\rightarrow \infty} = (\sqrt{2})^{e}$ and let $s_n= \sum_{k=1}^{n}a_n$. If $\{x_n\}$ is a strictly decreasing sequence of real numbers with $\lim_{n\rightarrow \infty} x_n = (e)^{\sqrt2}$, then find the value of $$\lim_{n\rightarrow \infty}\frac{1}{s_n}\sum_{k=1}^{n}a_kx_k$$

I tried to calculate. Will it converge to $(\sqrt{2})^{e}(e)^{\sqrt2}$?

Answer 1

We have the following case of Cesaro-Stolz theorem:

• $s_n = \sum_{k=1}^{n}a_n \nearrow +\infty$
• $\lim_{n\rightarrow\infty}\frac{a_n x_n}{a_n} = \lim_{n\rightarrow\infty}x_n = e^{\sqrt{2}}$

$$\Rightarrow \lim_{n\rightarrow \infty}\frac{1}{s_n}\sum_{k=1}^{n}a_kx_k = e^{\sqrt{2}}$$