Infinite sums in python

Question

I have heard that python can do infinite sums. For instance if I want to evaluate the infinite sum:

1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + ...

How should I go about? I am a newbie to python. So I would appreciate if someone could write out the entire code and if I need to include/import something.

For instance, in wolfram alpha if I input Summation (-1)^(n-1)/(2*n-1) from n=1 to infinity it gives the answer as 0.785395. I want the answer computed to a desired accuracy say, as in the case of wolfram alpha upto 6 digits.

Further, I was looking at this post here and tried to mimic that but it gives me the following errors:

`NameError: name 'Infinity' is not defined`
`NameError: name 'Inf' is not defined`

Thanks, Adhvaitha

Answer 1

While it still is finite, you can approximate that series using the fractions and decimal modules:

from fractions import Fraction
from decimal import Decimal

repetitions = 100

d = 1
r = Fraction(1, d)

for n in range(repetitions):
    r += Fraction(1, d) - Fraction(1, d + 2)
    d += 4

print(Decimal(r.numerator)/Decimal(r.denominator))

I think this comes closest to what you want to do.

Answer 2

Python has unlimited precision integers, but not unlimited precision floats. There are packages you can use that provide that, though.

And nothing can "complete" an infinite sum, since it involves an infinite number of steps. You'll need to find a closed form for the sum, and then evaluate that, or accept an approximation achieved by terminating the infinite sum when a precision criterion is met.