How to use Continued-fraction in a sentence
He was also the first to consider the difficult problems involved in equations of mixed differences, and to prove that an equation in finite differences of the first degree and the second order might always be converted into a continued fraction.
The exact determination of the appearances in any given case is a mere problem of convergents to a continued fraction.
If the continued fraction terminates, it is said to be a terminating continued fraction; if the number of the quantities a, ..., b 2 ..
If b 2 /a 2, 3 /a 3 ..., the component fractions, as they are called, recur, either from the commencement or from some fixed term, the continued fraction is said to be recurring or periodic. It is obvious that every terminating continued fraction reduces to a commensurable number.
The notation employed by English writers for the general continued fraction is al b2 b3 b4 a 2 "' Continental writers frequently use the notation a 1 ?Advertisement
The general continued fraction al is evidently equal, convergent by convergent, to the continued fraction X 2 b 2 X2X3b3 x3%4b4 a1+ A2a2 + X + X
The simple continued fraction is both the most interesting and important kind of continued fraction.
A non-terminating simple continued fraction must be incommensurable.
In the case of a terminating simple continued fraction the number of partial quotients may be odd or even as we please by writing the I last partial quotient, a n as a n - I +1.
The difference between the continued fraction and the nth convergent is less than, and greater than a n+2 These limits qn may be replaced by the following, which, though not so close, are simpler, viz.Advertisement
Every simple continued fraction must converge to a definite limit; for its value lies between that of the first and second convergents and, since f ?n _ _1 I, L t.
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 b4 as a4 b5 approximation.
Thus a continued fraction equivalent to 7r (the ratio of the circumference to the diameter of a circle) is I I I I II 3+ 7+15+7+292+i-1-i+ ..
Similarly the continued fraction given by Euler as equivalent to 1(e - 1) (e being the base of Napierian logarithms), viz.
If we suppose alb 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.Advertisement
The second case illustrates a feature of the recurring continued fraction which represents a complete quadratic surd.
In the case of a recurring continued fraction which represents N, where N is an integer, if n is the number of partial quotients in the recurring cycle, and pnr/gnr the nr th convergent, then p 2 nr - Ng2nr = (- I) nr, whence, if n is odd, integral solutions of the indeterminate equation x 2 - Ny 2 = I (the so-called Pellian equation) can be found.
The theory and development of the simple recurring continued fraction is due to Lagrange.
It is always possible to find the value of the n th convergent to a recurring continued fraction.
We have seen that the simple infinite continued fraction converges.Advertisement
The infinite general continued fraction of the first class cannot diverge for its value lies between that of its first two convergents.
The infinite continued fraction a2 +a3+ an ..
A continued fraction may always be found whose n th convergent shall be equal to the sum to n terms of a given series or the product to n factors of a given continued product.
In fact, a continued fraction ai +a2+ +an+ can be constructed having for the numerators of its successive convergents any assigned quantities pi, P2, P3,
If we form then the continued fraction inwhich pi, p2, p3 9 ..., pn are u1, u1 + u 2, ul+u2+u3,Advertisement
There is, however, a different way in which a series may be represented by a continued fraction.
Lambert for expressing as a continued fraction of the preceding type the quotient of two convergent power series.
There is another type of continued fraction called the ascending continued fraction, the type so far discussed being called the descending continued fraction.
The theory, however, starts with the publication in 16J5 by Lord Brouncker of the continued fraction I 23252 i 2 + 2 + 2 +.
Huygens (Descriptio automati planetarii, 1703) uses the simple continued fraction for the purpose of approximation when designing the toothed wheels of his Planetarium.
For the convergence of the continued fraction of the second class there is no complete criterion.