The Chudnovsky series is based on a hypergeometric series, which may be why you think it is expressible as a simple geometric series. However, in general hypergeometric series are not expressible as geometric series.
However, using the expression you've linked to at wikipedia, you can write a trivial implementation in Python:
So we have:
For some large value of $x$. So we can write the following:
# x is the limit of the summation, increase the value of x
# for a more accurate approximation.
sum = 0
while x >= 0:
sum += (((-1)**x) * factorial(6*x) * (13591409 + 545140134*k))/(factorial(3*k)*(factorial(k)**3) * (640320**(3*x + (3/2))))
x -= 1
sum *= 12
However, bear in mind it's been a while since I've done Python scripting, this script was made just based on documentation I could find on the Python site, so I'm unsure if the syntax/semantics are correct, but the concept is there.
Hope this helps.