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limit of a sequence

Source: IMC 1997 day 1 problem 1

October 1, 2005
limitlogarithmscalculusintegrationreal analysisreal analysis unsolved

Problem Statement

Let {ϵn}n=1∞\{\epsilon_n\}^\infty_{n=1} be a sequence of positive reals with lim⁡n→+∞ϵn=0\lim\limits_{n\rightarrow+\infty}\epsilon_n = 0. Find lim⁡n→∞1n∑k=1nln⁡(kn+ϵn) \lim\limits_{n\rightarrow\infty}\dfrac{1}{n}\sum\limits^{n}_{k=1}\ln\left(\dfrac{k}{n}+\epsilon_n\right)
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