1 a_{n+2}
is less than ------------, and greater than ------------. These limits
q_{n}q_{n+1} q_{n}q_{n+2}
may be replaced by the following, which, though not so close, are
1 1
simpler, viz. ------- and ------------------ .
q^{2}_n q_n(q_n + q_{n+1})
Every simple continued fraction must converge to a definite limit; for
its value lies between that of the first and second convergents and,
since
p_n p_{n-1} 1 p_n p_{n-1}
--- ~ ------- = ------------, Lt. ----- = Lt. -------,
q_n q_{n-1} q_{n}q_{n-1} q_n q_{n-1}
so that its value cannot oscillate.
The chief practical use of the simple continued fraction is that by
means of it we can obtain rational fractions which approximate to any
quantity, and we can also estimate the error of our approximation. Thus
a continued fraction equivalent to [pi] (the ratio of the circumference
to the diameter of a circle) is
1 1 1 1 1 1
3 + - -- -- --- -- --
7 + 15 + 1 + 292 + 1 + 1 + ...
of which the successive convergents are
3 22 333 355 103993
--, --, ---, ---, ------, &c.,
1 7 106 113 33102
the fourth of which is accurate to the sixth decimal place, since the
error lies between 1/q4q5 or .0000002673 and a6/q4q6 or .0000002665.
Similarly the continued fraction given by Euler as equivalent to
1/2(e -1) (e being the base of Napierian logarithms), viz.
1 1 1 1 1
-- -- -- -- --
1 + 6 + 10 + 14 + 18 + ...,
may be used to approximate very rapidly to the value of e.
For the application of continued fractions to the problem "To find the
fraction, whose denominator does not exceed a given integer D, which
shall most closely approximate (by excess or defect, as may be assigned)
to a given number commensurable or incommensurable," the reader is
referred to G. Chrystal's _Algebra_, where also may be found details of
the application of continued fractions to such interesting and important
problems as the recurrence of eclipses and the rectification of the
calendar (q.v.).
Lagrange used simple continued fractions to approximate to the solutions
of numerical equations; thus, if an equation has a root between two
integers a and a + 1, put x = a + 1/y and form the equation in y; if the
equation in y has a root between b and b + 1, put y = b + 1/z, and so
on. Such a method is, however, too tedious, compared with such a method
as Homer's, to be of any practical value.
The solution in integers of the indeterminate equation ax + by = c may
be effected by means of continued fractions. If we suppose a/b to be
converted into a continued fraction and p/q to be the penultimate
convergent, we have aq - bp = +1 or -1, according as the number of
convergents is even or odd, which we can take them to be as we please.
If we take aq-bp = +1 we have a general solution in integers of ax + by
= c, viz. x = cq - bt, y = at - cp; if we take aq - bp = -1, we have x =
bt - cq, y = cp - at.
An interesting application of continued fractions to establish a unique
correspondence between the elements of an aggregate of m dimensions and
an aggregate of n dimensions is given by G. Cantor in vol. 2 of the
_Acta Mathematica_.
Applications of simple continued fractions to the theory of numbers, as,
for example, to prove the theorem that a divisor of the sum of two
squares is itself the sum of two squares, may be found in J. A. Serret's
_Cours d'Algebre Superieure_.
2. _Recurring Simple Continued Fractions._--The infinite continued
fraction
1 1 1 1 1 1 1 1 1 1
a1 + -- -- --- -- -- --- -- -- --- --
a2 + a3 ... + a_n + b1 + b2 ... + b_n + b1 + b2 ... + b_n + b1 + ...,
where, after the n^{th} partial quotient, the cycle of partial quotients
b1, b2, ..., b_n recur in the same order, is the type of a recurring
simple continued fraction.
The value of such a fraction is the positive root of a quadratic
equation whose coefficients are real and of which one root is negative.
Since the fraction is infinite it cannot be commensurable and therefore
its value is a quadratic surd number. Conversely every positive
quadratic surd number, when expressed as a simple continued fraction,
will give rise to a recurring fraction. Thus
__ 1 1 1 1 1
2 - \/ 3 = -- -- -- -- --
3 + 1 + 2 + 1 + 2 + ...,
___ 1 1 1 1 1 1 1 1
\/ 28 = 5 + -- -- -- -- -- -- -- --
3 + 2 + 3 + 10 + 3 + 2 + 3 + 10 + ...
The second case illustrates a feature of the recurring continued
fraction which represents a complete quadratic surd. There is only one
non-recurring partial quotient a1. If b1, b2, ..., b_n is the cycle of
recurring quotients, then b_n = 2a1, b1 = b_{n-1}, b2 = b_{n-2}, b3 =
b_{n-3}, &c.
In the case of a recurring continued fraction which represents [sqr]N,
where N is an integer, if n is the number of partial quotients in the
recurring cycle, and p_{nr}/q_{nr} the nr^{th} convergent, then p^2_{nr}
-Nq^2_{nr} = (-1)^{nr}, whence, if n is odd, integral solutions of the
indeterminate equation x squared - Ny squared = +-1 (the so-called Pellian equation)
can be found. If n is even, solutions of the equation x squared -Ny squared = +1 can
be found.
The theory and development of the simple recurring continued fraction is
due to Lagrange. For proofs of the theorems here stated and for
applications to the more general indeterminate equation x squared -Ny squared = H the
reader may consult Chrystal's _Algebra_ or Serret's _Cours d'Algebre
Superieure_; he may also profitably consult a tract by T. Muir, _The
Expression of a Quadratic Surd as a Continued Fraction_ (Glasgow, 1874).
_The General Continued Fraction._
1. _The Evaluation of Continued Fractions._--The numerators and
denominators of the convergents to the general continued fraction both
satisfy the difference equation u_n = a_{n}u_{n-1} + b_{n}u_{n-2}. When
we can solve this equation we have an expression for the n^{th}
convergent to the fraction, generally in the form of the quotient of two
series, each of n terms. As an example, take the fraction (known as
Brouncker's fraction, after Lord Brouncker)
1 1 squared 3 squared 5 squared 7 squared
-- -- -- -- --
1 + 2 + 2 + 2 + 2 + ...