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Posted by Peter Munn on February 4, 2008, 3:49 pm
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Leafing through alt.sci.planetary, I read David Williams's message of
Thu, 10 Jan 2008:
>-> > Wrong series. The total volume would be 1/1^2+1/2^2+1/3^2... which is
>-> > 1+1/4+1/9+1/16... This series may also have a finite sum, but off the
>-> > top of my head I can't think of a way to prove it.
>
>I tried a brute force and ignorance approach, just making a computer
>calculate the sum of many terms. Because of the square, the terms
>become very small quite fast. The sum of a million terms is only very
>slightly greater than the sum of 100,000 terms, so I suspect that the
>infinite series does have a finite sum, which turns out to be (if I
>remember it right) about 1.645. That's a bit bigger than pi/2, and a
>bit smaller than sqr(e). I don't know if there is any simple expression
>that would define it exactly.
Looking it up in David Wells' _Dictionary of Curious and Interesting
Numbers_, I see it's (pi^2)/6.
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