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Find the limit

Source: 2019 Jozsef Wildt International Math Competition-W. 8

May 18, 2020
limitSummationHarmonic NumbersSequencesgreatest integer funtion

Problem Statement

Let (an)n≥1(a_n)_{n\geq 1} be a positive real sequence given by an=∑k=1n1ka_n=\sum \limits_{k=1}^n \frac{1}{k}. Compute lim⁡n→∞e−2an∑k=1n⌊(k!2k+(k+1)!2(k+1))2⌋\lim \limits_{n \to \infty}e^{-2a_n} \sum \limits_{k=1}^n \left \lfloor \left(\sqrt[2k]{k!}+\sqrt[2(k+1)]{(k+1)!}\right)^2 \right \rfloorwhere we denote by ⌊x⌋\lfloor x\rfloor the integer part of xx.
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