Page:Elementary algebra (1896).djvu/424

406 Rh Cor. 2. The difference between two successive convergents is a fraction whose numerator is unity, and whose denominator is the product of the denominators of these convergents ; for

506. Each convergent is nearer to the continued fraction than any of the preceding convergents.

Let x denote the continued fraction, and $$\frac{p_{n}}{q_{n}}, \frac{p_{n+1}}{q_{n+1}}, \frac{p_{n+2}}{q_{n+2}}$$ three consecutive convergents; then $$x$$ differs from $$\frac{p_{n+2}}{q_{n+2}}$$ only in taking the complete (n + 2)th quotient in the place of $$a_{n+2}$$; denote this by $$k$$; thus

Now k is greater than unity, and $$q_n$$ is less than $$q_{n+1}$$; hence on both accounts the difference between $$\frac{p_{n+1}}{q_{n+1}}$$ and $$x$$ is less than the difference between $$\frac{p_{n}}{q_{n}}$$ and $$x$$; that is, every convergent is nearer to the continued fraction than the next preceding convergent, and therefore nearer than any preceding convergent.

Combining the result of this article with that of Art. 501, it follows that

The convergents of an odd order continually increase, but are always less than the continued fraction ;

The convergents of an even order continually decrease, but are always greater than the continued fraction.


 * The sign ~ means “ difference between.”