I was recently told to compute some integral, and the result turned out to be a scalar multiple of the series ∞∑n=11(n(n+1))p, where p≥1. I know it converges by comparison for 1(n(n+1))p≤1n(n+1)<1n2, and we know thanks to Euler that ∞∑n=11n2=π26. I managed to work out the cases where p=1 and p=2. With p=1 being a telescoping sum, and my solution for p=2 being 13π2−3, which I obtained based on Euler's solution to the Basel Problem. I see no way to generalize the results to values to arbitrary values of p however. Any advice on where to start would be much appreciated.
Also, in absence of another formula, is the series itself a valid answer? Given that it converges of course.
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