Question

Newton-Raphson method for solving algebraic equations is based on

• A. Taylor series
• B. Fourier series
• C. Laurent series
• D. Power series

Hint:

Think about how a tangent line built from a Taylor expansion truncated after the linear term gives the next guess.

Answer 1

The Correct Option is A

Step 1: Picture the tangent line at the current guess.
At a point \( x_0 \), draw the tangent line to the curve \( y = f(x) \); its slope is \( f'(x_0) \).

Step 2: Write the tangent line equation.
\[ y - f(x_0) = f'(x_0)(x - x_0) \]
The next guess \( x_1 \) is taken where this tangent line crosses \( y = 0 \).

Step 3: Solve for the crossing point.
\[ 0 - f(x_0) = f'(x_0)(x_1 - x_0) \implies x_1 = x_0 - \frac{f(x_0)}{f'(x_0)} \]
This is exactly the Newton-Raphson update rule.

Step 4: Connect the tangent line back to its source.
A tangent line at a point is nothing but the first order (linear) truncation of the Taylor series of \( f(x) \) about that point.

Final Answer:
The method reduces to using a truncated Taylor series at each step, not a Fourier, Laurent, or general power series. \[ x_1 = x_0 - \dfrac{f(x_0)}{f'(x_0)} \]