Finding numerical approximations to π has fascinated people for millenia. One famous formula is
Say 
is the sum of the first k terms of the series above, and 
. Here is a fancy way to compute these sequences in a compact code.
is the sum of the first k terms of the series above, and . Here is a fancy way to compute these sequences in a compact code.k = (1:100)';
s = cumsum( 1./k.^2 ); % cumulative summation
p = sqrt(6*s);
plot(k,p,'.-')
xlabel('k'), ylabel('p_k')
title('Sequence converging to \pi')
This graph suggests that 
but doesn't give much information about the rate of convergence. Let 
be the sequence of errors. By plotting the error sequence on a log-log scale, we can see a nearly linear relationship.
but doesn't give much information about the rate of convergence. Let be the sequence of errors. By plotting the error sequence on a log-log scale, we can see a nearly linear relationship.ep = abs(pi-p); % error sequence
loglog(k,ep,'.'), title('log-log convergence')
xlabel('k'), ylabel('error'), axis tight
This suggests a power-law relationship where 
, or 
.
, or .V = [ k.^0, log(k) ]; % fitting matrix
c = V \ log(ep) % coefficients of linear fit
In terms of the parameters a and b used above, we have
a = exp(c(1)), b = c(2),
It's tempting to conjecture that 
asymptotically. Here is how the numerical fit compares to the original convergence curve.
asymptotically. Here is how the numerical fit compares to the original convergence curve. hold on, loglog(k,a*k.^b,'r'), title('power-law fit')
axis tight % ignore this line