Arthur Ernest Jolliffe

“ 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’Algèbre Supérieure. ”

“ The sum of an infinite series may be deduced from the sum to rt terms, when this is known, by increasing n indefinitely and finding the limit, if any, to which it tends, but a series may often be summed to infinity when it cannot be summed to n terms ”

“ If ui, u¢, are functions of a variable z, a convergent infinite product (I -l-u1) (I -I-uz) . defines a function of z. For such products there is a theory of uniform convergence analogous to that of infinite series. Is is not in general possible to represent a function as an infinite product; the question has been dealt with by Weierstrass (see his Abhandlungen aus der Functionlehre or A. R. Forsyth's Theory of Functions) . One of the simplest cases of a function expressed as an infinite product is that of sin z/z, which is the value of the absolutely convergent infinite product. ”

“ The Conversion of Series and Products into Continued Fractions. 1. A continued fraction may always be found whose nth 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
can be constructed having for the numerators of its successive convergents any assigned quantities p1, p2, p3, . . ., pn, and for their denominators any assigned quantities q1, q2, q3, . . ”

“ If 'X/u, . approaches a limit l as n is indefinitely increased, Eu.. will converge if l is less than unity and will diverge if is greater than unity (Cauchy's test) ; if u, ,+1/u, . approaches a limit l as n is indefinitely increased, Eu.. will converge if l is less than unity and diverge if l is greater than unity (D'Alembert's test) . Nothing is settled when the limit l is unity, except in the case when remains greater than unity as it approaches unity. The series then diverges. ”

“ 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/qnr the nrth convergent, then p2nr −Nq2nr = (−1) nr, whence, if n is odd, integral solutions of the indeterminate equation x2 − Ny2 = ±1 (the so-called Pellian equation) can be found. If n is even, solutions of the equation x2 −Ny2 = +1 can be found. ”

“ In mathematics a set of quantities, real or complex, arranged in order so that each quantity is definitely and uniquely determined by its position, is said to form a series. Usually a series proceeds in one direction and the successive terms are denoted by ”

“ John Wallis, discussing this fraction in his Arithmetica Infinitorum (1656) , gives many of the elementary properties of the convergents to the general continued fraction, including the rule for their formation. Huygens (Descriptio automati planetarii, 1703) uses the simple continued fraction for the purpose of approximation when designing the toothed wheels of his Planetarium. Nicol Saunderson (1682–1739) , Euler and Lambert helped in developing the theory, and much was done by Lagrange in his additions to the French edition of Euler’s Algebra (1795) . ”

“ E. Goursat, Cours d’analyse (translated by E. R. Hedrick) , vol. i. (1902) ; J. Harkness and F. Morley, A Treatise on the Theory of Functions (1893) and Introduction to the Theory of Analytic Functions (1899) ”