Monday, 3 July 2017

calculus - Prove suminftym=1suminftyn=1frac1n(m2+n2)=suminftym=1suminftyn=1frac1n2(m2+n2)=fracpi472



How may I prove that
m=1n=11n(m2+n2)=m=1n=11n2(m2+n2)=π472?
I also discussed the problem in the chat, but no solution so far. Some hints? Thanks!


Answer



For now, here is how we can prove the second equality. Let the second sum be S. Note that by symmetry we also have S=m=1n=11m2(m2+n2). Now adding the two forms gives: 2S=m=1n=11m2n2=(m=11m2)(n=11n2)=π436.




As Fabian alludes to in the comments, it appears the first equality does not hold, since the difference between the two sums is m=1n=1n2nn31(m2+n2)>0.


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