Arthur Ernest Jolliffe

“ An infinite series may contain both positive and negative terms. The terms may be positive and negative alternately or they may occur in groups which without altering the order of the terms of the series may each be collected into a single term; thus all series may be regarded as belonging to one of two types, ul-i-u2+u3+ in which the terms are all positive, or u1-u2-|-u3- in which the terms are alternately positive and negative. 8. It is clear that if a series is convergent un must tend to the limit zero as n is increased indefinitely. ”

“ A power series converges absolutely and uniformly at every point within its circle of convergence; it may be differentiated or integrated term by term; the function represented by a power 'series is continuous within its circle of convergence and, if the series is convergent on the circle of convergence, the continuity extends on to the circle of convergence. Two power series cannot be equal throughout any region in which both are convergent Without being identical. ”

“ Hence Σ (δu/δx1) δx1 must vanish for all sets of increments δx1, . . . δxn, and since these are independent, we must have δu/δx1 = 0, δu/δx2 = 0, . . . δu/δxn = 0. A value of u given by a set of solutions of these equations is called a “critical value” of u. The value of δu now becomes
for u to be a maximum or minimum this must have always the same sign. ”

“ 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. ”

“ Some problems may be solved by algebra; thus if y = ƒ (x) ÷ φ (x) , where ƒ (x) and φ (x) are polynomials in x, the limits to the values of yφ may be found from the consideration that the equation yφ (x) − ƒ (x) = 0 must have real roots. This is a useful method in the case in which φ (x) and ƒ (x) are quadratics, but scarcely ever in any other case. The problem of finding the maximum product of n positive quantities whose sum is given may also be found, algebraically, thus. ”

“ 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, . . ”

“ For the case of a single variable x, corresponding to a value of x given by the equation du/dx = 0, u is a maximum or minimum as d 2u/dx2 is negative or positive. If d 2u/dx2 vanishes, then there is no maximum or minimum unless d 2u/dx2 vanishes, and there is a maximum or minimum according as d 4u/dx4 is negative or positive. Generally, if the first differential coefficient which does not vanish is even, there is a maximum or minimum according as this is negative or positive. If it is odd, there is no maximum or minimum. ”

“ 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. ”

“ 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. ”

“ 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 ”

“ 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. ”

“ Of two regular polygons of the same perimeter, that with the greater number of sides encloses the greater area.
3. The circle encloses a greater area than any polygon of the same perimeter.
4. The sum of the areas of two isosceles triangles on given bases, the sum of whose perimeters is given, is greatest when the triangles are similar.
5. Of segments of a circle of given perimeter, the semicircle encloses the greatest area.
6. The sphere is the surface of given area which encloses the greatest volume. ”

“ 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) ”