Discussion
The real sequence generated by the iterative scheme
\[ x_{n} = \frac{x_{n-1}}{2} + \frac{1}{x_{n-1}},\ n \geq 1 \]
Answer: B
The sequence converges to \(\sqrt{2}\) whenever \(x_{0} \gt \sqrt{2/3}\).
The scheme is Newton's method for \(x^{2} - 2 = 0\): \(x - \frac{x^{2}-2}{2x} = \frac{x}{2} + \frac{1}{x}\). For any \(x_{0} \gt 0\), the AM-GM inequality gives
\[ \begin{aligned} x_{1} &= \frac{x_{0}}{2} + \frac{1}{x_{0}} \\ &\geq 2\sqrt{\frac{1}{2}} = \sqrt{2} \end{aligned} \]
From then on the terms stay at or above \(\sqrt{2}\) and decrease, so they converge to the positive fixed point \(\sqrt{2}\). Every \(x_{0} \gt \sqrt{2/3}\) is positive, so the statement holds.
The map is odd, so a negative start converges to \(-\sqrt{2}\). That makes 'for all \(x_{0} \neq 0\)' false, and also the (-1, 1) option, which includes negative starts. The sequence does not diverge for any non-zero start.
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