Yes, computers may be used to calculate p with millions of digits accuracy but this requires special effort. Actually the computer represents numbers with a finite "string" of 0 and 1. If the difference between two numbers is small the computer may not be able to distinguish them.

Experiment with the following algorithm:

Accu=1
Repeat until there is no difference between 1 and 1+Accu
   Accu=Accu/2
   Check if (1+Accu=1)

Try the same excersice in: (a) EXCEL, (b)  Maple and (c) if you can on FORTRAN.