convergents to a continued fraction
For , define
The convergent to is given by
where are defined as above.
Induction. For ,
For , the numbers and are a Farey pair; in fact,
This is again a simple induction. The statement is true for . For , we have
Note that if is a simple continued fraction, then the above theorem implies that , since any common factor of and must divide .
Similar to the proof of the above theorem. ∎
If is a simple continued fraction, then and, for , .
This follows directly from the iterative definition for the and the fact that the are positive integers. ∎
These results easily imply the following important convergence theorem:
For any continued fraction, the even convergents are strictly monotonically increasing, and the odd convergents are strictly monotonically decreasing. In addition, every odd convergent is greater than each even convergent. If the continued fraction is simple, then the limit of the odd convergents is equal to the limit of the even convergents, and thus the continued fraction has a well-defined value equal to their common limit.
This is basically obvious from the previous observations. Write for the convergent, i.e.
Each is positive, so
is positive for even and negative for odd. This proves the observations about monotonicity. Also,
is positive for odd, so that
Next we prove the following theorem regarding the connection between the “tail” of a continued fraction, its convergents, and its value:
If is a simple continued fraction, write for (the complete convergent). Then
This is another simple proof by induction. Note that
Finally, we derive a bound on how well the convergents approximate the value of the continued fraction:
If is a simple continued fraction, then
But , so that and thus
since the are strictly increasing. ∎
- 1 G.H. Hardy & E.M. Wright, An Introduction to the Theory of Numbers, Fifth Edition, Oxford Science Publications, 1979.
|Title||convergents to a continued fraction|
|Date of creation||2013-03-22 18:04:20|
|Last modified on||2013-03-22 18:04:20|
|Last modified by||rm50 (10146)|