InfSumPy 1.0.1

Approximate infinite sums with a guaranteed error.

---

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

Instalation

Make sure you have the mpmath library installed:

pip install mpmath

To install the package, run the following command:

pip install infsumpy

Usage

We have the transformations implemented above, and for use 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 espected error tolerance (if method is ratio, integral or threshold).
• L (optional): Limit of the ratio of terms (if method is ratio).
• Integral of series (optional): The function of g(n) = ∫_n^∞ f(x) dx for the inetegral test (if method is integral).
• Precision (optional): The precision for the mpmath library (default value if 53).

The function return 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)