Sums of Reciprocal Powers

You have seen that the  diverges. What about the sum of reciprocal squares? In fact, they converge, and to something very interesting:

SUMk=1 to infinity ( 1/k2 ) = ( 2/6 )

Where did that Pi come from, anyway?

If you liked that one, here are more:

SUMk=1 to infinity ( 1/k4 ) = ( Pi4/90 ) 
SUMk=1 to infinity ( 1/k6 ) = ( Pi6/945 ) 
SUMk=1 to infinity ( 1/k8 ) = ( Pi8/9450 ) 
SUMk=1 to infinity ( 1/k10 ) = ( Pi10/93,555 )

Presentation Suggestions:
This Fun Fact is short and fun for the class to ponder.

The Math Behind the Fact:
Little is known about sums of odd powers. It was recently shown (Apery) that the sum of the cubed reciprocals is . The sums of reciprocal powers as you vary the power is a function known as the .

How to Cite this Page: 
Su, Francis E., et al. “Sums of Reciprocal Powers.” Math Fun Facts. <https://www.math.hmc.edu/funfacts>.

References:
Math. Intelligencer article by van der Poorten, 78/79 MR 80i:10054

Fun Fact suggested by:
Lesley Ward

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