Calculate Nested Radical to 10 Decimal Places in Python

Program Overview

This Python program calculates the value of the infinite nested radical expression:
$$\sqrt{n + \sqrt{n + \sqrt{n + \cdots}}}$$
The calculation continues until the difference between two consecutive steps is less than 1e-10, ensuring 10-digit precision.

Python Code:

import math

def nested_sqrt(n: float, precision: int = 10) -> float:
    epsilon = 10 ** (-precision)
    current = math.sqrt(n)
    while True:
        next_val = math.sqrt(n + current)
        if abs(next_val - current) < epsilon:
            break
        current = next_val
    return round(current, precision)

# Read input from user
n = float(input("Enter a number n: "))
result = nested_sqrt(n)
print(f"Value of the nested radical for n = {n} is: {result}")

Sample Output:

Input: 5  
Value of the nested radical for n = 5 is: 3.3301910412

Step-by-Step Explanation:

- The initial value is set to √n
- Each iteration computes √(n + previous value)
- The loop stops when the change between steps is less than 10⁻¹⁰
- The final result is rounded to 10 decimal places and displayed