Approximate π Using a Custom Fractional Series in Python

Program Overview

This Python program approximates the value of π using a custom fractional series:
$$\frac{\pi}{2} = \frac{2}{1} + \frac{2}{3} + \frac{4}{3} + \frac{4}{5} + \frac{6}{5} + \frac{6}{7} + \cdots$$
The pattern alternates between repeating numerators and increasing odd denominators.
The user inputs the number of terms n to compute.

Python Code:

def approximate_pi(n):
    total = 0
    num = 2
    den = 1

    for i in range(n):
        total += num / den
        if i % 2 == 1:
            num += 2
        den += 2

    pi_estimate = total * 2
    return round(pi_estimate, 6)

# Run the program
n = int(input("Enter number of terms: "))
result = approximate_pi(n)
print(f"Approximate value of π: {result}")

Sample Output:

Enter number of terms: 10  
Approximate value of π: 3.142857

Step-by-Step Explanation:

- Start with numerator = 2 and denominator = 1
- Add each term: numerator / denominator
- Every two terms, increase numerator by 2
- Denominator increases by 2 each time
- Multiply the final sum by 2 to approximate π