Let $x_1 = 100, $and for n ≥ 1, let $ x_{n+1}=\frac12(x_n +\frac{100}{x_n}).$ Assume that $L = lim_{n→∞}x_n$ exists, and calculate L.
Answer provided is 10. But I don't understand how it was calculated?
Let $x_1 = 100, $and for n ≥ 1, let $ x_{n+1}=\frac12(x_n +\frac{100}{x_n}).$ Assume that $L = lim_{n→∞}x_n$ exists, and calculate L.
Answer provided is 10. But I don't understand how it was calculated?
$x_{n+1} = \dfrac{1}{2}\left( x_n + \dfrac{100}{x_n} \right)$
Take limit of both sides as $n \to \infty$:
$\displaystyle \begin{align*}\lim_{n \to \infty} x_{n+1} & = \lim_{n \to \infty} \dfrac{1}{2} \left( x_n + \dfrac{100}{x_n} \right) \\ L & = \dfrac{1}{2}\left( \lim_{n \to \infty} x_n + \dfrac{100}{\displaystyle \lim_{n \to \infty} x_n } \right) \\ L & = \dfrac{1}{2}\left( L + \dfrac{100}{L} \right)\end{align*}$
Solve for L.