Nth digit of PI

[Andrew Dalke]

Randall Hopper
>But one of the interesting things that you can show with a Fourier
>integral (possibly other ways too) is that:
>
>    Pi^2/8 = sum(n=1..inf, 1/(2n-1)^2)
>
>You can see that the sum is 1/1 + 1/9 + 1/125 + ....  The terms get smaller
>and smaller, which means making the decimal representation longer and
>longer.  In the limit, the decimal representation is infinitely long
>because the denominator is infinitely large.


I've seen the proof that e is irrational, but don't recall the details.
The sketch you outlined isn't sufficient.  Consider

  sum(n=1..inf, 1/n^2) = 1 + 1/2 + 1/4 + 1/8 + ... = 2

For any finite N in the sum 1..N, the denominator is infinitely large,
but the infinite sum is an integer.

                    Andrew
                    dalke at acm.org