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- #1

- Feb 5, 2012

- 1,621

Can somebody give me a hint to solve this problem.

**Problem:**

Let \(f\) be a function defined on \([a,\,b]\) with continuous second order derivative. Let \(x_0\in (a,\,b)\) satisfy \(f(x_0)=0\) but \(f'(x_0)\neq 0\). Prove that, there is a neighbourhood of \(x_0\), say \(U(x_0)\), such that, for all \(x_1\in U(x_0)\), the following sequence,

\[x_{n+1}=x_{n}-\frac{f(x_n)}{f'(x_n)}\]

where \(n=1,\,2,\,\cdots\) is convergent.