You don't need the series form for pi.
I'll show you how to do it for e = 2.7818... and you can see for yourself how
to do it for pi.
The integer part of e is 2, so here A=2, B=e-2.
Thus f(e) = A + (1-B) = 2 + (1-(e-2)) = 5-e=2.21...
(and yes, I'm the Calkin in Calkin-Wilf-Newman).