INFINITESIMAL CALCULUS 19 44. We may observe that Taylor also introduced into his Methodus Incrementorum, in the fluxional notation, a series which is the same as that of Bernoulli, already noticed. This led to a long and bitter controversy between them, in which Bernoulli s son Nicholas and others also took part. In this Taylor was accused of plagiarism both with respect to this theorem and to other theorems relative to the general theory of the centre of oscillation of bodies. It is re markable that in this dispute no reference was made to Taylor s own theorem, nor do the disputants seem to have been aware of its vast superiority to that around which the angry controversy was raised. 45. Taylor s theorem seems never to have risen into due promi nence until its great value was pointed out by the illustrious Lagrange, in the Berlin memoirs for 1772. Lagrange demonstrated the theorem by the principles of ordinary algebra. He made it the foundation of the method of series, and also of the differential calculus. He thus proposed to make the calculus a branch of ordinary algebra, and independent of all considerations of infinitely small quantities, and so to give it all the formal rigour of demon stration of the method of the ancients. 46. Lagrange also was the first to place Taylor s theorem on a satisfactory basis by finding an expression for the remainder of the series after any number of terms. The following demonstration of this theorem of Lagrange depends on a single lemma, which may be thus stated. If a continuous /unction f(x) vanish when x = a, and also when x = b, then its derived function f (x), if also continuous, must also vanish for some value of X between a and b. This is easily proved ; for if f (x] does not T anish for some value of x between a and b, it must have always the same sign between these limits, and consequently/if*) must constantly increase or constantly diminish as x passes by small increments from the value a to the value b ; but this is impossible, since/ (x) vanishes for both limits. Now let E n represent the remainder after n terms in Taylor s expan sion, then writing X for x + y in that series, we have in which f(x), f (x) f( n ~ l x) are supposed finite and con tinuous for all vaiues of the variable between X and x. From the form of the terms included in E it evidently may be written in the shape R _(X-s)"p where P is some function of X and x. Consequently we have /(X)- /(a) + *)/ (d)+ ... + (X ~? l 1/( " 1)(cg) EL P =0 Now, let z be substituted for a: in every term in the preceding, with the exception of P, and let F(z) represent the resulting expres sion, we shall have in which P has the same value as before. Again, the right-hand side in this equation vanishes when z = X, . . F(X) = 0. Also, from (#), the right-hand side vanishes when z=x; Accordingly, since the function F(s) vanishes when =X, and also when z = 3 , it follows from the preceding lemma that its derived function F (z) also vanishes for some value of z between the limits X and x. Proceeding to obtain F (?) by differentiation, it can be easily seen from equation (7) that we have Consequently, for some value of 2 between a; and X we must have /<>(*) r P. Again, if be a positive quantity less than unity, the expression . + 0(X-#), by assigning a suitable value to 0, can be made eqiial to any number intermediate between x and X. Hence P =/(){ -Pfl(X-a)}, where 6 is some quantity >0 and <1. Consequently, the remainder after n terms of Taylor s series can be represented by This is Lagrange s form for the remainder. Substituting this value for E in (a), it becomes /(X)=/(,.) + ^/(*) + ^^/V) + . . . Again, if h be substituted for X - JL; the series becomes f(x + h) =/(*) + J,f(x) + & e . + ^li /(- i)( a .) + ^/() ( M + eh). In this expression n may be any positive integer. 47. The last equation may be regarded as the most general form of Taylor s theorem. We infer from it that the essential conditions for the application of Taylor s theorem to the expansion of any function in a series are that none of its derived functions should h n become infinite, and that ,--/() (ce + 0/t) should become infinitely small when n becomes sufficiently large. 48. The remainder in Taylor s series admits, as was shown by Cauchy, of being written in the form Another form was given by Dr Schlomilch, viz. K =Erff?r>><*+A>. In some cases one or other of these latter values is preferable to Lagrange s form. 49. Another remarkable mode of determining the remainder in Taylor s theorem was also given by Cauchy. It is based on the following lemma, that if ~F(x) and/(x) be two functions which re main continuous, as also their derived functions, between the values Xi and x l + h of x, and if also f (x) does not become zero for any value of x between these limits, then = f(x l + h)-f(x 1 ) / ( where 6 is less than unity. 50. If in Taylor s series we make x + h=Q, or h= -x, we get /(O) and hence a result which can be readily identified with Bernoulli s series, given in 38. 51. Again, ifa; = 0, Taylor s series becomes or, as it may be written, /(*)=/(<>) +y/ (0) + ~ in which f(Q), / (O), /"(O), &c., represent the values of f(x), f(x), f"(x), &c., when x = 0. This result is usually called Maclaurin s series, having been given in his Fluxions (1742). It had, however, been previously published by Stirling in his Mcth. Diff. (1717) ; but neither Stirling nor Maclaurin laid any claim to the theorem as being original, both referring it to Taylor. _ By substituting forf(x) any of the elementary functions, such as sin x, cos x, log (1+x), we readily obtain their well-known expan sions. It is to be noted that it is necessary in each case, for the validity of the series, to show that the remainder after n terms be comes indefinitely small when n is taken sufficiently large. 52. The application of Taylor s or of Maclaurin s theorem becomes extremely troublesome in many cases, owing to the complexity of the successive derived functions. For example, if we seek to expand tan x by Maclaurin s theorem, we have f(x) = tan x, f (x) = sec 2 x, f"(x) = 2 sec 2 a; tan x, f "(x) = 2 sec 4 + 4 sec 2 o; tan 2 x- ; and the subsequent derived functions increase in complexity. Similarly in the case of other elementary functions, such as sec ;>, cot x, &c. 53. The development of tail X, sec x, and many other functions is much facilitated by the aid of a system of numbers, introduced by James Bernoulli. These numbers are usually arrived at as fol lows. It is easilv seen that the expansion of :>l . . in ascending e*-l powers of x, contains no odd power of x after the first, and that the two first terms of the expansion are 1 and - . Accordingly
we may assume x
