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The Newton-Raphson method is a root-finding algorithm that uses the idea of approximating a function with a tangent line to iteratively refine an estimate of a root. Starting with an initial guess, the method repeatedly applies a formula to get successively better approximations until a desired level of accuracy is reached. 

Proof:

The method is based on the Taylor series expansion of a function f(x) around a point x₀. Truncating the series after the linear term, we get:

f(x) ≈ f(x₀) + f'(x₀)(x - x₀) 


We want to find the root, so we set f(x) = 0 and solve for x (which we'll call x₁ for the next approximation):

0 = f(x₀) + f'(x₀)(x₁ - x₀) 

Rearranging the equation, we get the Newton-Raphson formula:

x₁ = x₀ - f(x₀) / f'(x₀) 


Formula:

The general formula for the Newton-Raphson method is:
xᵢ₊₁ = xᵢ - f(xᵢ) / f'(xᵢ) 
where:
xᵢ is the current approximation of the root.
xᵢ₊₁ is the next, improved approximation.
f(xᵢ) is the value of the function at xᵢ.
f'(xᵢ) is the value of the first derivative of the function at xᵢ. 


Example:
The Newton-Raphson method is a root-finding algorithm that uses the idea of approximating a function with a tangent line to iteratively refine an estimate of a root. Starting with an initial guess, the method repeatedly applies a formula to get successively better approximations until a desired level of accuracy is reached. [1, 2]  

Proof: 
The method is based on the Taylor series expansion of a function f(x) around a point x₀. Truncating the series after the linear term, we get: [2, 3, 4]  
f(x) ≈ f(x₀) + f'(x₀)(x - x₀) [2, 3, 4]  

We want to find the root, so we set f(x) = 0 and solve for x (which we'll call x₁ for the next approximation): [2, 3, 4, 4]  
0 = f(x₀) + f'(x₀)(x₁ - x₀) [3, 4]  

Rearranging the equation, we get the Newton-Raphson formula: 
x₁ = x₀ - f(x₀) / f'(x₀) [3, 4]  

Formula: 
The general formula for the Newton-Raphson method is: 
xᵢ₊₁ = xᵢ - f(xᵢ) / f'(xᵢ) [3, 5]  

where: 

• xᵢ is the current approximation of the root. 
• xᵢ₊₁ is the next, improved approximation. 
• f(xᵢ) is the value of the function at xᵢ. 
• f'(xᵢ) is the value of the first derivative of the function at xᵢ. [3, 3, 4, 4, 5, 5]  


Example: 

Let's find the root of f(x) = x² - 2 (which is √2). [2, 3, 3, 5, 5, 6, 7]  

1. Choose an initial guess: Let's start with x₀ = 1. 

2. Calculate the derivative: f'(x) = 2x 

3. First iteration: 
	• f(x₀) = f(1) = 1² - 2 = -1 
	• f'(x₀) = f'(1) = 2 * 1 = 2 
	• x₁ = 1 - (-1 / 2) = 1.5 

4. Second iteration: 
	• f(x₁) = f(1.5) = 1.5² - 2 = 0.25 
	• f'(x₁) = f'(1.5) = 2 * 1.5 = 3 
	• x₂ = 1.5 - (0.25 / 3) = 1.41666... 

5. Third iteration: 
	• f(x₂) = f(1.41666...) ≈ -0.0069 
	• f'(x₂) = f'(1.41666...) ≈ 2.8333 
	• x₃ = 1.41666... - (-0.0069 / 2.8333) ≈ 1.414215 

After a few iterations, we get a value very close to √2 ≈ 1.414213. [3, 5]

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