Sunday, February 24, 2019

real analysis - Compute sumlimitsin=1nftyfrac1(n(n+1))p where pgeq1

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 p1. I know it converges by comparison for 1(n(n+1))p1n(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 13π23, 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.

No comments:

Post a Comment

analysis - Injection, making bijection

I have injection f:AB and I want to get bijection. Can I just resting codomain to f(A)? I know that every function i...