An Intermediate Result

To apply the extrapolation procedure to the sequence of tangent approximations we now simply call Richardson(piTangent,5,0,16,2). This says: we want to compute 5 rows, all columns (0), we want compute with a precision of 16 digits, and info level 2, indicating that we want the extrapolation triangle displayed.

r := Richardson(piTangent,5,0,16,2):
evalf(r,11),evalf(Pi,11),evalf(Pi-r,16);

  i  T[i]

  0 +3.2000000000 

  1 +3.1623529412 +3.1498039216 

  2 +3.1468005184 +3.1416163775 +3.1410705412 

  3 +3.1428947296 +3.1415928000 +3.1415912282 +3.1415994930 

  4 +3.1419181743 +3.1415926559 +3.1415926463 +3.1415926688 +3.1415926420 

  5 +3.1416740338 +3.1415926536 +3.1415926535 +3.1415926536 +3.1415926535 +3.1415926535 

Richardson(piTangent,...) gains approximately 4 digits with each iteration. This is quiet nice. But we can do a little better.