Tuesday, 28 April 2015

sequences and series - Mathematical Explanation of Mathematica Summation suminftyn=1frac(2n1)!(2n+2)!zeta(2n)


From a mathematical point of view, what phenomena that most likely Mathematica Wolfram encountered when calculating:
n=1(2n1)!(2n+2)!ζ(2n)=2log(2π)38+ζ(3)8π2


which is incorrect.




While calculating the sum from this question, I noticed that Wolfram result is containing ζ(3)8π2, which is incorrect. Although I realized that this could be a bug, I started to wonder if there are any logical explanation behind this miscalculation! Has Wolfram algorithm encountered something similar to Riemann Rearrangement Theorem?



Doing more investigations, it turns-out that Wolfram is incorrectly miscalculating the closed form of an entire class of zeta summation, except the last case which is correct.

n=1ζ(αn)(n+a)(n+b)=n=1[Aζ(αn)n+a+Bζ(αn)n+b+]=C+n=1ζ(αn)1(n+a)(n+b)=n=1[Aζ(αn)1n+a+Bζ(αn)1n+b+]+C


And with the appearance of this case (the last correct closed form), I believe there is a mathematical explanation regarding a correct summation method or algorithm that gives a kind of systematic incorrect closed form if it applied in a certain way. Appreciating if someone can explore this and alert us regardless of any bug that may exist in any math app. Thanks.



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