I may in passing observe that this discriminating quantity may be expressed in the form d. log f (x) (26) d. log x In this form it was first given by Cauchy. If the series, whose character is to be determined, is a f (x), then, if we replace f (2) the preceding quantity becomes - *}{[%); and consequently the discriminating quantity of La f(x) is – (27) f(x) and the series is convergent when this quantity is greater than 1, and is divergent when it is less than 1. The quantity (27) may also be put into the form d.log f(x) (28) On
Ex. 1. Shew that the series il is convergent. If f(x) = x (x+1); then " -= 2, when x = 0. f(x) x2 + x Consequently the series is convergent. Ex. 2. Determine the character as to convergence or divergence of the series (x + 1)a 1 za+b xato Here from (25), if f(x) = (x+1)a' x f'(20) 6x2 +(a+b)x = b, if x = 0. f(x)= x2 + x Consequently the series is convergent if 6 is greater than 1, and divergent if b is less than 1. Ex. 3. Es va com is convergent when a is greater than 1, and is divergent when a is less than 1. Ex. 4. 2 is divergent. 186.] If the discriminating quantity determined in the last article=1, when x = 00; the test fails, and the series may thus far be either convergent or divergent. In this case let us compare the given series with • {x (log x)"}-1, the character of which is determined in Ex. 3, Art. 183, and which is therein shewn to be convergent or divergent according as m is greater than or less than 1. In this case f'(x) m-, when x = 0; (log 2)" + m (log x)-1' = a finite quantity, suppose, when x = ; so that f'(x) logic (m + log x) f (x) :: m= log « * * 1) - 1}, when x = c. (29) which is the required discriminating quantity; and the series 3 convergent or divergent according as this quantity is greater than or less than 1. Ex. l. Determine the character as to convergence and divergence of the series i (at –1). In this case the discriminating quantity aš log a _1? = log x3 8 lxaš –1) = 0, when x = 00: consequently the series is divergent. Ex. 2. The series 2 t1 is divergent. 187.] If however the discriminating quantity given in (29) is equal to 1, the test fails, and the series may thus far be either convergent or divergent. In this case we may compare the series with : {x log x (log? x)"}-1, which has been proved to be convergent or divergent according as m is greater than or less than 1; and comparing Lotion with this series, and making it comparable with it, we find as in the last two articles, and the given series is convergent or divergent according as m is greater than or less than 1. If the value of m given in (30) = 1, we must proceed in a similar manner to another of the series given in Ex. 5, Art 183 : and so on. Hereby we obtain the following series of discriminating quantities, viz. w f'(x), Saf'(x) f(x) I f (2) .. when x = 0; (31) of which the law of formation is evidently Qn = logM–1 x {Qn-1-1}, (32) where Q1 = (33) 188.] Another series of discriminating quantities has been derived by Raabe from the preceding; and in certain forms of functions these are applied with greater facility. Take (1) to be the form of the series ; then since Uz Up -1 = f(x+1)-f(x) f(x) and thus C may be replaced by -1. Hence Q1, which is given in (33), = x } (34) and the discriminating quantities given in (31) and (32) become changed accordingly. This latter form of the discriminating quantity is convenient, when a series of factorials enters into the general term of the series, as the following examples will shew. Ex. l. Determine the character as to convergence and divergence of X 1.3.5 ... (2x-1) 1 2.4.6 ... 2 x 2x+1 Ug+ 'S= (2x + 1)2 = 5, when x = 00; hence the series is convergent. ma 1.3.5 ... (2x – 1). Ex. 2. The series 4 is divergent. 2.4.6 ... 2 x Another form has been given to the preceding theorems by M. Paucker of St. Petersbourg*: but as they are exactly equivalent, it is unnecessary to insert them. 189.] All the preceding investigations, as to the convergence and divergence of series, may be summed up in the following practical directions to the student. Examine first the ratio of Uz+1 to u, when x = 0; according as that ratio is greater than or less than 1, the series is divergent or convergent. If that ratio = 1, then examine in order the series of discriminating quantities given in (31), or in the equivalent forms as modified by (34); then according as that discriminating quantity, which is the first not to be equal to 1 when x = 0, is greater or less than 1, so will the series be convergent or divergent. I cannot conclude this section without acknowledging my obligations to Mr. De Morgan, M. Duhamel, and H. Raabe; but above all to the critical Memoir of M. J. Bertrand on the convergence of series, in Vol. VII, of Liouville's Journal, in which he compares the several criteria of the three mentioned writers, and demonstrates the identity of them, and consequently shews that the extent of applicability is the same in all. I would also refer the reader to two Memoirs on the Theory of Series by Ossian Bonnet. The former on Convergence and Divergence, given in Liouville, Vol. VIII, p. 73; the latter in Mém. Cour. de l'Acad. Roy. de Belgique, Tome XXIII, in which he treats especially of Periodic Series. SECTION 2.--The Series of Taylor and Maclaurin. 190.] In the first article of the preceding section, it has been remarked that there are two cases in which an exact equality exists between a function and the series into which it is developed; and consequently in which these quantities may be employed interchangeably with each other. These cases are (1) that in which the series is convergent, and the number of terms is infinite; (2) that in which the remainder, that is the sum of all the terms after the nth, can be determined, and added to the preceding n terms. It has also been observed that a series may be convergent for certain values of its variable and divergent for other values. Now in the proofs of the theorems of Taylor and Maclaurin, which are given in Chapter VI. of Vol. I, it is shewn that * See Crelle's Journal. Vol. XLII, p. 138. PRICE, VOL. II. K k (36) (37) the functions which are to be expanded must in the first place be such, that neither they nor their derived functions up to the (n-1)th inclusively must be infinite within certain limits, when the development is effected as far as the nth term. This condition is previous to any test which determines the convergence or divergence of the series; this latter character must be determined by the criteria of the preceding section. If Taylor's Theorem is given in the form f(a+h) = f(a)+F(a)i + ... + F*-1(a) 1234 In_+ ..., (35) then taking the discriminating quantity in the form (34), we have (72-1 (a) x (36) 2 * (a) and consequently, the series (35) is convergent or divergent, according as (36), when x = oo, is greater than or less than 1. Again, taking Maclaurin's series in the form hx-1 F (h) = F(0) +F(0) 4 + ... +p*-1(0)123 this series will be convergent or divergent according as x { F*-?(0) *. - 1}, when x = 0, F? (0) h 1 is greater than or less than 1. 191.] In the preceding volume the remainders of Taylor's and Maclaurin's series have also been investigated; but the forms are indeterminate, because they depend on 0 which is an undetermined positive proper fraction. But the remainders can be found in more precise forms as definite integrals. These we proceed to determine; and the inquiry is also otherwise important, because it yields new proofs of these theorems, subject however to the same conditions, as to continuity and finiteness of the function and the derived functions, as those which are required in the previous proofs. Let f'(x+h-2) be a function, which is finite and continuous for all employed values of its subject-variable ; let it be the element-function of a z-integration, and let h and 0 be the limits of z; then *t'(x+h–z)dz = [ -f(x+h–z)]. (39) = f(x+h)-f(x). (40) (38) |