The -th Harmonic number, denoted is defined by the sum of the reciprocals of the first  natural numbers:     This number has quite interesting properties. On thing we could consider when is finite is the quotient of the sum. If we write it as     Then many problems raise. Here, we have a problem concerning when is the denominator the lcm of . 2022 AMC 12A Problem 23 Let…

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Let be a continuous integrable real function on , then     Solution 1 By Cauchy-Schwarz inequality,     Solution 2 By Jensen’s inequality, since is a convex function, we have     So     Solution 3 By Hölder’s Inequality, we get     Choose , we get the case of solution 1. We can also take and , which would be similar. Solution 4

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