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Prove that the sequence is convergent and find its limit

Source: 2019 Jozsef Wildt International Math Competition

May 20, 2020

Sequenceslimitintegrationcalculus

Problem Statement

Let be

x_1=\frac{1}{\sqrt[n+1]{n!}}

and

x_2=\frac{1}{\sqrt[n+1]{(n-1)!}}

for all

n\in \mathbb{N}^*

and

f:\left(\left .\frac{1}{\sqrt[n+1]{(n+1)!}},1\right.\right] \to \mathbb{R}

where

f(x)=\frac{n+1}{x\ln (n+1)!+(n+1)\ln \left(x^x\right)}

Prove that the sequence

(a_n)_{n\geq1}

when

a_n=\int \limits_{x_1}^{x_2}f(x)dx

is convergent and compute

\lim \limits_{n \to \infty}a_n