Question

Show by the theory of continued fractions that $\frac{99}{70}$ differs from $1\cdot 41421$ by a quantity less than $\frac{1}{11830}$.

Answer 1

To show by the theory of continued fractions that $\frac{99}{70}$ differs from $1.41421$ by a quantity less than $\frac{1}{11830}$

We have,

$1.41421=1+\frac{1}{2+}\frac{1}{2+}\frac{1}{2+}\frac{1}{2+}\frac{1}{2+}\frac{1}{2+}.........$ ;

The convergents are

$1,\frac{3}{2},\frac{7}{5},\frac{17}{12},\frac{41}{29},\frac{99}{70},\frac{239}{169}.........$;

$\therefore$ the error in taking $\frac{99}{70}$ as an Approximation $<\frac{1}{70\times 169}=\frac{1}{11830}$