Question

Express the following quantities as continued fractions and find the fourth convergent to each
$\cdot 3029$

Answer 1

To Express the following quantities as continued fractions and to find the fourth convergent to each .

$0.3029$

$=\frac{3029}{10000}$

This can be written as

$1+\frac{139}{1000}$

$\therefore$ the Successive quotients are $3,3,3,6,1,2,1,10$

and This can be written as Continued fraction

$0.3029=1+\frac{1}{3+}\frac{1}{3+}\frac{1}{3+}\frac{1}{6+}\frac{1}{1+}\frac{1}{2+}\frac{1}{1+}\frac{1}{10};$

and the first four convergents are $\frac{1}{3},\frac{3}{10},\frac{10}{33},\frac{63}{208}$

Hence , fourth Convergent is $\frac{63}{208}$