500. Convergents. The fractions obtained by stopping at the first, second, third, quotients of a continued fraction are called the first, second, third, convergents, because, as will be shown in Art. 506, each successive convergent is a nearer approximation to the true value of the continued fraction than any of the preceding convergents.
501. To show that the convergents are alternately less and greater than the continued fraction.
Let the continued fraction be a_1 + {1}{a_2 +} {1}{a_3 +}
The first convergent is a_, and is too small because the part {1}{a_2 +} {1}{a_3 +} is omitted. The second convergent is a_1 + {1}{a_2 +}, and is too great because the denominator a_2, is too small. The third convergent is a_1 + {1}{a_2 +} {1}{a_3} and is too small because is a_2 + {1}{a_3} too great; and so on.
When the given fraction is a proper fraction, a_1= 0; if in this case we agree to consider zero as the first convergent, we may enunciate the above results as follows:
The convergents of an odd order are all less, and the convergents of an even order are all greater, than the continued fraction.
502. To establish the law of formation of the successive convergents.
Let the continued fraction be denoted by
a_1 + {1}{a_2 +} {1}{a_3 +} {1}{a_4 +} ;
then the first three convergents are
{a_1}{1}, {a_1a_2+1}{a_2}, {a_3(a_1a_2+1)+a_1}{a_3a_2+1};
