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Mathematik-Online problems:

Problem 95: Monotony, Boundedness, Convergence of a Recursive Defined Sequences

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
#./aufgabe95_en.tex#Let $ a$ be a positive number and $ (x_n)$ the recursive defined sequence

$\displaystyle x_0=a,\quad x_{n+1} = \frac{1}{2}\left(x_n+\frac{a}{x_n}\right),\quad n\in\mathbb{N}_0. $

a)
Analyse if $ (x_n)$ is monotone respectively bounded.
b)
Show that $ \vert x_{n+1}-\sqrt a\,\vert = \frac{1}{2x_n}\,\vert x_n-\sqrt a\,\vert^2$, for all $ n\in\mathbb{N}_0$, and use this to proof the convergence of the sequence $ (x_n)$ against $ \sqrt{a}$.
c)
Estimate the error $ \vert x_{20}-\sqrt{a}\vert$ for $ a=2$.

(Authors: Höllig/Höfert)

see also:

[Solutions]
  automatisch erstellt am 6.  2. 2018