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Find the limit of the expression made by 3 sequences

Source: 2019 Jozsef Wildt International Math Competition-W. 11

May 18, 2020
Sequenceslimit

Problem Statement

Let (sn)n≥1(s_n)_{n\geq 1} be a sequence given by sn=−2n+∑k=1n1ks_n=-2\sqrt{n}+\sum \limits_{k=1}^n\frac{1}{\sqrt{k}} with lim⁡n→∞sn=s=\lim \limits_{n \to \infty}s_n=s=Ioachimescu constant and (an)n≥1(a_n)_{n\geq 1} , (bn)n≥1(b_n)_{n\geq 1} be a positive real sequences such that lim⁡n→∞an+1nan=a∈R+∗,lim⁡n→∞bn+1bnn=b∈R+∗\lim \limits_{n\to \infty}\frac{a_{n+1}}{na_n}=a\in \mathbb{R}^*_+, \lim \limits_{n\to \infty}\frac{b_{n+1}}{b_n\sqrt{n}}=b\in \mathbb{R}^*_+Computelim⁡n→∞(1+esn−esn+1)anbnn\lim \limits_{n\to \infty}\left(1+e^{s_n}-e^{s_{n+1}}\right)^{\sqrt[n]{a_nb_n}}
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