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

Newton-Raphson is a root-finding method, and the question asks what mathematical idea it is built on.

Start from a Taylor series expansion of a function $f(x)$ around a current guess $x_0$: $f(x) \approx f(x_0) + f'(x_0)(x - x_0)$, keeping only the constant and linear terms and dropping everything of higher order. Setting this linear approximation to zero and solving for $x$ gives the next guess, $x_1 = x_0 - f(x_0)/f'(x_0)$, which is exactly the Newton-Raphson update formula.

1. Taylor series: correct, the method comes directly from truncating a Taylor expansion at the first order term.
2. Fourier series: wrong, that series represents a periodic function as a sum of sines and cosines, unrelated to local root finding.
3. Laurent series: wrong, that expansion handles complex functions with singularities, not this iterative formula.
4. Power series: wrong, a general power series is not what produces the specific linear update step used here, the Taylor expansion at a point is.

So Newton-Raphson is based on the Taylor series, option (A).