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405
ALGEBRA
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405
We have seen that
Let the continued fraction be denoted by x; then x differs
from only in taking the complete quotient k instead of the partial quotient ; thus
505. To show that if 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 , 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 and had a common divisor it would divide or unity ; which is impossible.