A cartoon of the Stolz–Cesàro Theorem // La caricatura di Teorema Stolz-Cesàro

 In the Mathematical Review of Students from Timișoara, RMT for short In the Revista Matematică a Elevilor din Timișoara, RMT for short, I found a problem (page 69) that I solved in my time :

 "3304.  Given the sequences  $\{a_n\}_{n\in\mathbb{N}}\;,\;\;\{b_n\}_{n\in\mathbb{N}}$ with the properties :

 (i)  $\{b_n\}_{n\in\mathbb{N}}$  is strictly monotone and unbounded.

                               (ii)  exist  $\displaystyle \lim_{n\to \infty}\frac{a_n}{b_n}$

                               (iii)  $\frac{a_{n+1}}{a_n}+\frac{b_{n+1}}{b_n}=2\;,\;\;(\forall) n\in\mathbb{N}.$

                           Prove that  $\displaystyle \lim_{n \to \infty}\frac{a_n}{b_n}=0.$

{author :} Titu ANDREESCU, student, Timișoara"

Solution CiP

              The condition  (iii)  is written equivalently : $\frac{a_{n+1}}{a_n}-1+\frac{b_{n+1}}{b_n}-1=0\;\Leftrightarrow$

(to continue)

Stolz–Cesàro theorem