MathDB

Too easy limit of sequence of mean values

Source: Romanian National Olympiad 2016, grade xii. p.1

August 25, 2019

real analysislimits

Problem Statement

Prove that there exists an unique sequence

\left( c_n \right)_{n\ge 1}

of real numbers from the interval

(0,1)

such that

\int_0^1 \frac{dx}{1+x^m} =\frac{1}{1+c_m^m } ,

for all natural numbers

m,

and calculate

\lim_{k\to\infty } kc_k^k.

Radu Pop