Encyclopaedia Britannica, 11th Edition, "Constantine Pavlovich" to "Convention" / Volume 7, Slice 2

by Various

In the case of a terminating simple continued fraction the number of partial quotients may be odd or even as we please by writing the last partial quotient, a_n as

an - 1 +	1		.
	1

The numerators and denominators of the successive convergents obey the law p_nq_n-1 - p_n-1q_n = (-1)ⁿ, from which it follows at once that every convergent is in its lowest terms. The other principal properties of the convergents are:—

The odd convergents form an increasing series of rational fractions continually approaching to the value of the whole continued fraction; the even convergents form a decreasing series having the same property.

Every even convergent is greater than every odd convergent; every odd convergent is less than, and every even convergent greater than, any following convergent.

Every convergent is nearer to the value of the whole fraction than any preceding convergent.

Every convergent is a nearer approximation to the value of the whole fraction than any fraction whose denominator is less than that of the convergent.

The difference between the continued fraction and the n^th convergent is

less than	1	, and greater than	an+2	.
	qnqn+1		qnqn+2

These limits may be replaced by the following, which, though not so close, are simpler, viz.