Page:Elementary algebra (1896).djvu/423

We have seen that

Let the continued fraction be denoted by x; then x differs

from ${\frac {p_{n}}{q_{n}}}$ only in taking the complete quotient k instead of the partial quotient $a_{n}$; thus

505. To show that if ${\frac {p_{n}}{q_{n}}}$ be the nth convergent to a continued fraction, then

Let the continued fraction be denoted by

then similarly,

But ; hence .

When the continued fraction is less than unity, this result will still hold if we suppose that $a_{1}=0$, and that the first convergent is zero.

Note. When we are calculating the numerical value of the successive convergents, the above theorem furnishes an easy test of the accuracy of the work.

Cor. 1. Each convergent is in its lowest terms; for if $p_{n}$ and $q_{n}$ had a common divisor it would divide $p_{n}q_{n-1}-p_{n-1}q_{n}$ or unity ; which is impossible.