and the first of these which tends, as x is indefinitely increased to a limit greater or less than unity, determines the series to be convergent or divergent. The criteria may be presented in another form. For 1 representing – by $ (w), and applying to each of the functions in (A), the rule for indeterminate functions of the form = Ø - 1} - 1}, ($ (oc) 1) 1 lla φ (α) x logo (oc) = log ac $() and so on. Thus the system of functions (A) is replaced by the system a¢' (20) fæ¢' (x) la 6' (c) &c. , (B). • (C) It was virtually under this form that the system of functions was originally presented by Prof. De Morgan, (Differential Calculus, pp. 325-7). The law of formation is as follows. If P, represent the nth function, then Pote = 1"» (P.» – 1). ...... (11). 10. There exists yet another and equivalent system of determining functions which in particular cases possesses great advantages over the two above noted. It is obtained by sub Uz φ' (α) stituting in Prof. De Morgan's forms 1 for The Φ (α) lawfulness of this substitution may be established as follows. Ust Uz = we have Uxto 1 Since $(x) $ (2) $ (2) φ() $(x) $ (2+0) .(12). Now $'(x+) has unity for its limiting value; for, 4 (2) φ' (α) tends to become infinite as ac is indefinitely increased, and $(x+) therefore assumes the form 20; therefore $ (2) $ (x + ) _$ (+0) φ' (α) $ (2) and $(x+1); or in other words tends to φ (α) φ (α) the limit 1. Thus (12) becomes $'(x) 1 $ (20) Substituting therefore in (B), we obtain the system of functions $(x + 1), i.e. 1 and Uxt? the law of formation being still Patz = 7"x (P-1). 11. The extension of the theory of degree referred to in Art, 5 is involved in the demonstration of the above criteria. When two functions of a are, in the ordinary sense of the term, of the same degree, i.e. when they respectively involve the same highest powers of x, they tend, x being indefinitely increased, to a ratio which is finite yet not equal to 0; viz. to the ratio of the respective coefficients of that highest power. Now let the converse of this proposition be assumed as the definition of equality of degree, i.e. let any two functions of x be said to be of the same degree when the ratio between them tends, x being indefinitely increased, to a finite limit which is not equal to 0. Then are the several functions X (7.C)", xlx (11x)", &c., 1 with which or $ (m) is successively compared in the de Ux monstrations of the successive criteria, so many interpositions of degree between x and alta, however small a may be. For x being indefinitely increased, we have (2.c) 2 (1.c) lim lim -0, lo (II.)" æla (lla)* lim lim ali = 0, 2 (lx)ita so that, according to the definition, a (1x)" is intermediate in degree between 2 and achta, xlx (llx)" between xlx and ac (bachata, &c. And thus each failing case, arising from the supposition of m=1, is met by the introduction of a new function. It may be noted in conclusion that the first criterion of the system (A) was originally demonstrated by Cauchy, and the first of the system (C) by Raabe (Crelle, Vol. 1x.). Bertrand *, to whom the comparison of the three systems is due, has demonstrated that if one of the criteria should fail from the absence of a definite limit, the succeeding criteria will also fail in the same way. The possibility of their continued failure through the continụed reproduction of the defivite limit 1, is a question which has indeed been noticed but has scarcely been discussed. * Liouville's Journal, Tom. VII. p. 35. m 1ta 12. The results of the above inquiry may be collected into the following rule. RULE. Determine first the limiting value of the function Ux+1 1. According as this is less or greater than unity the series Une is convergent or divergent. But if that limiting value be unity, seek the limiting values of whichsoever is most convenient of the three systems of functions (A), (B), (C). According as, in the system chosen, the first function whose limiting value is not unity, assumes a limiting value greater or less than unity, the series is convergent or divergent. Ex. 5. Let the given series be 21771 (x + 1)** (x+1)+ and x being indefinitely increased the limiting value is unity. Now applying the first criterion of the system (A), we have 1 7 l.c X + 1 ! u 3 + 1 and the limiting value is again unity. Applying the second criterion in (A), we have 1 1 2 ruir lloc 7x2 llic Tac allic the limiting value of which found in the usual way is 0. Hence the series is divergent. a a (a + 1)(a +2) 4+&c...(14), Ex. 6. Resuming the hypergeometrical series of Ex. 4, viz. a (a +1) 1+ ++ ť + b (b + 1) b (b + 1) (6+2) we have in the case of failure when t= 1, a (a + 1) ... (a +* — 1) = (b-a) x atx which tends to the limit b - a. The series is therefore convergent or divergent according as b-a is greater or less than unity. If b - a is equal to unity, we have, by the second criterion of (C), Uz - a). s(6 lac loc a +30 - alac ata since b - a=1. The limiting value is 0, so that the series is still divergent. It appears, therefore, 1st, that the series (14) is convergent or divergent according as t is less or greater than 1; 2ndly, that if t =1 the series is convergent if b-a>1, divergent if b - a=1. |