Page:Elementary algebra (1896).djvu/427

409 Rh 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. $$\frac{253}{179}$$. 5. $$\frac{251}{802}$$. 6. $$\frac{1189}{3927}$$. 7. $$\frac{729}{2318}$$. 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 $$1 + \frac{1}{3 +} \frac{1}{5 +} \frac{1}{7 +} \frac{1}{9 +} \frac{1}{11 +} \ldots$$ which differs from the true value by less than .0001.

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

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.