Summary

We considered the equation

Int((1-t)/((t+1)*sqrt(t)),t = 0 .. 1) = 2/3*hypergeom([1, 2],[5/2],1/2);

The left hand side led us to the identity

'2*I*ln((1-I)/(1+I))'=2*I*ln((1-I)/(1+I));

which is equivalent to Euler's famous formula , and led us further to propose the extrapolation method

for numerical evaluation of Pi. But then it became clear, that an efficient computation of

'pi(x)' = 'sum(2^(x+2)/((n+1/2+2^x)^2+(2^x)^2),n=0..infinity)';

would pay our bill. This point is still under consideration and you are invited to contribute.

When we turned to the right hand side, we were led to the identity

'sum(2^(m+1)/(m*binomial(2*m,m)),m=1..infinity)' = sum(2^(m+1)/(m*binomial(2*m,m)),m=1..infinity);

This in turn is the starting point of the spigot algorithm for the computation of Pi, which amounts to a base conversion from a mixed-radix base to some base.

So now Euler's equation turned into a limiting relation

'limit(sum(2^(x+2)/((n-1/2+2^x)^2+(2^x)^2),n=1..infinity),x=infinity)'='sum(2^(m+1)/(m*binomial(2*m,m)),m=1..infinity)';

or, in our new notation, things end happily as  .