InfSumPy is a Python package that evaluates infinite positive sums with guaranteed error. Using ratio and integral tests we evaluate series that pass these tests with controlled error.

Instalation

To install the package, run the following command:

pip install infsumpy

Usage

We have the transformations implemented above, and for use, we have the infsum function. Which receives from input:

• A series: In the form of a function f: $\mathbb{N} \to \mathbb{R}$.
• Method: Can be ratio, integral, threshold or fixed.
• Max terms: The maximum number of terms.
• Start terms: The index of the first term of the series.
• Epsilon (optional): The expected error tolerance (if the method is ratio, integral or threshold).
• L (optional): Limit of the ratio of terms (if the method is ratio).
• Integral of series (optional): The function of g(n) = ∫_n^∞ f(x) dx for the integral test (if the method is integral).
• Precision (optional): The precision for the mpmath library (default value is 53).

The function returns the number of terms used in the sum and the approximation.

Ratio test

from infsumpy import infsum

# the infinity sum of n/(2**n) pass in the ratio test with limit L = 1/2,
# then we can evaluate with controled error
print(infsum(lambda n: n/(2**n), 'ratio', max_terms=10**4, initial=1, eps=2**(-52), L=1/2))

> (56, 2.0)

Integral test

from infsumpy import infsum

# the infinity sum of 1/n**2 pass in the integral test with integral
# g(n) = ∫_n^∞ 1/x**2 dx = 1/n, then we can evaluate with controled error
print(infsum(lambda n: 1/(n**2), 'integral', max_terms=10**4, initial=1, eps=10**(-3), g=lambda n: 1/n))

> (499, 1.64493406229104)

Threshold (not guaranteed)

from infsumpy import infsum

# we can also use a stoping criterio such that sum until the n-th are less
# than the epsilon, here for the infinity sum of 2/(2**n)
print(infsum(lambda n: n/(2**n), 'threshold', max_terms=10**4, initial=1, eps=2**(-52)))

> (57, 2.0)

Fixed (not guaranteed)

from infsumpy import infsum

# we can just sum a fixed number of terms of the infinite sum of 2/(2**n)
print(infsum(lambda n: n/(2**n), 'fixed', max_terms=10**4, initial=1))

> (10000, 2.0)