EXAMPLES XLIII. a.
Calculate the successive convergents to
1. 2. 3.
Express the following quantities as continued fractions and find the fourth convergent to each : also determine the limits to the error made by taking the third convergent for the fraction.
4. . 5. . 6. . 7. . 8. .37. 9. 1.139. 10. .3029. 11. 4.316.
12. Find limits to the error in taking \frac{222}{203} yards as equivalent to a metre, given that a metre is equal to 1.0936 yards.
13. Find an approximation to which differs from the true value by less than .0001.
14. Show by the theory of continued fractions that differs from 1.41421 by a quantity less than .
RECURRING CONTINUED FRACTIONS.
511. We have seen that a terminating continued fraction with rational quotients can be reduced to an ordinary fraction with integral numerator and denominator, and therefore cannot be equal to a surd; but we shall prove that a quadratic surd can be expressed as an infinite continued fraction whose quotients recur. We shall first consider a numerical example.
Ex. Express 19 as a continued fraction, and find a series of fractions approximating to its value.
