Part CCSIR NET December 2025the-derivative-vanishing-at-the-root-too-signals-a-multiple-root-newtons-method-degrades-to-linear-convergence-there

The derivative vanishing at the root too signals a multiple root newtons method degrades to linear convergence there

Let { $x_{n}$ } be a convergent iterative sequence generated by Newton-Raphson method for solving the equation sin x−1=0 such that $x_{n}\to\pi/2$ as $n\to\infty$ . For $n\in\mathbb{N}$ , let $e_{n}=x_{n}-\pi/2$ . Let p>0 be such that $\lim(n\to\infty)$ | $e_{n}$ ₊ $_{1}$ |/| $e_{n}$ |^p exists and is non-zero. Then which of the following statements are true?

A.p = 1✓
B. $\lim(n\to\infty)$ | $e_{n}$ ₊ $_{1}$ |/| $e_{n}$ |^p = 1/2✓
C.p = 2
D. $\lim(n\to\infty)$ | $e_{n}$ ₊ $_{1}$ |/| $e_{n}$ |^p = 1

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward.

See pricing

50 are analysed free — try those first.

Related counterexample: Newton's method converges quadratically to any root

The derivative vanishing at the root too signals a multiple root newtons method degrades to linear convergence there

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