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--- pow:problem3s23 [2023/03/06 13:40] – created mazur
+++ pow:problem3s23 [2023/03/15 18:48] (current) – mazur
@@ Line 1: / Line 1: @@
+<box 85% round orange| Problem 3 (due Monday, March 6)>
+Let $\displaystyle s(n)=\sum_{j=1}^n {n\choose j} \frac{1}{j}$ and $\displaystyle f(n)=\frac{2^{n+1}}{n}$.
+Prove that
+\[\lim_{n\to\infty} n\left (\frac{s(n)}{f(n)}-1\right)\]
+exists and find its value.
+</box>
+The problem arose from a question my former PhD student Andrew Kelley asked me in December 2022.
+We received two solutions, from Prof. Vladislav Kargin and Prof. Anton Schick. Prof. Kargin's solution uses the central limit theorem
+and some careful estimates of the binomial coefficients. It is not hard to see that the central
+limit theorem is not really needed as the estimates of binomial coefficients are sufficient.
+Our original solution is more elementary.  Then, slightly after the "deadline", Prof. Anton Schick
+came with a fully probabilistic argument. For a detailed solution and additional results and problems
+see the following link {{:pow:2023sproblem3.pdf|Solution}}.