Stirling’s formula says that a good approximation to f(x) = x! (factorial)

Question

g(x) = √2πx (x/e)^x

where x! = x(x − 1)(x − 2)...3 · 2 · 1 whenever x is a positive integer, but it’s also defined for non-integers as well

a)The sense that g(x) is a “good” approximation of f(x) is that limx→∞g(x)/f(x) = 1

(b) Plot f(x) and g(x) on the same axes using the domain [0, 5] to further convince yourself

that this is a nice approximation.

Answer 1

To illustrate Stirling's approximation \( g(x) = \sqrt{2 \pi x} \left( \frac{x}{e} \right)^x \) compared to \( f(x) = x! \) over the domain [0, 5], we can plot both functions and observe their behavior.

Let's go through the steps to create this plot in Python using matplotlib:

```python

import math

import numpy as np

import matplotlib.pyplot as plt

# Function definitions

def factorial(x):

return math.factorial(x)

def stirling_approx(x):

return np.sqrt(2 * np.pi * x) * (x / np.e)**x

# Domain

x_values = np.linspace(0, 5, 100)

y_factorial = [factorial(int(x)) for x in x_values] # Calculate factorial for integer values of x

y_stirling = stirling_approx(x_values)

# Plotting

plt.figure(figsize=(10, 6))

plt.plot(x_values, y_factorial, label='x!', marker='o')

plt.plot(x_values, y_stirling, label='Stirling\'s Approximation', linestyle='--')

plt.title('Comparison of x! and Stirling\'s Approximation')

plt.xlabel('x')

plt.ylabel('Value')

plt.legend()

plt.grid(True)

plt.show()

```

### Explanation:

- **Factorial Function (`factorial(x)`)**: This uses Python's `math.factorial` function to calculate \( x! \).

- **Stirling Approximation Function (`stirling_approx(x)`)**: Implements Stirling's approximation formula for factorial.

- **Domain (`x_values`)**: Generates 100 points between 0 and 5 for plotting.

- **Plotting**:

- The `plot` function is used to plot both \( x! \) and Stirling's approximation \( g(x) \) against `x_values`.

- `marker='o'` and `linestyle='--'` are used for visual differentiation between the two lines.

- Axes labels, title, legend, and grid are added for clarity.

### Conclusion:

The plot will demonstrate that Stirling's approximation g(x) closely approximates the factorial function x! over the specified domain [0, 5], thereby supporting the statement that Stirling's formula provides a good approximation for factorial values, especially for large ( x ).