Calculate the Sum of a Fractional Product Series in Python

Program Overview

This Python program calculates the sum of the following infinite series:
$$\sum_{n=2}^{\infty} \left( \prod_{k=2}^{n} \frac{1}{k} \right)$$
Each term is a product of fractions starting from 1/2 and continuing multiplicatively.
The calculation continues until the next term becomes smaller than 1e-10.

Python Code:

def fractional_series_sum(threshold=1e-10):
    total = 0.0
    term = 1.0
    n = 2

    while True:
        term *= 1 / n
        if term < threshold:
            break
        total += term
        n += 1

    return round(total, 10)

# Run the program
result = fractional_series_sum()
print(f"Sum of the series: {result}")

Sample Output:

Sum of the series: 0.8269917904

Step-by-Step Explanation:

- The loop starts from n = 2
- Each term is calculated as a cumulative product: term *= 1/n
- If the term becomes smaller than 1e-10, the loop stops
- The final sum is rounded to 10 decimal places and displayed