f + j Ar_1 cos (r—1) <f>} mxm~l + kr~2 cos {nx+(r-2)$} m(m-l)*"-J + }■ If the function of which the general, say the rth, derivedfunction is to be found, is the quotient of one function of x by another function of x, so that m = -> the preceding formula (5) is applicable, if we take u to be one factor and i to be the other. The general formula for fr(x), as it stands, is however too complicated to be of any use. And the formula (5) is also indirectly applicable to the discovery of fr(x), even when the primitive function is not the product of two given functions of the variable x. The mode of its application will be best exhibited by examples. treatise we are not able to investigate all the conditions of such a possibility, yet doubtless the expansion is possible under the following circumstances. Suppose f(x) to be a function of x, and to be capable of expansion in a series of ascending powers of x, of the form f(x) — Ao + Ai# + A2#2 + ... + AnXn + ... (7) and suppose the series to be subject to the following conditions: 1st, A0, Ai, A2, ... A„, ... are constants, and to be determined. 2nd, Ao, Ai, A2,... A„, ... do not become infinite for any value of x for which the series is applied. 3rd, The series contains no terms involving negative or fractional powers of x. Then, since f(x) = Ao + Aitf-I- A2X2 + ... + A„Xn + ... • •• /'(*) = A1+^a2x + 3asx2+ ... This series was discovered by Stirling, an English Mathematician of the early part of the last century; but having been introduced by Maclaurin into his Treatise of Fluxions, has been generally called by his name. 58.] For certain functions of x, such as those of the form (a-f x)n, where n is positive and integral, the derived-functions will vanish after a certain number of differentiations; and therefore the number of terms of the above series is limited in such cases. Generally the successively derived-functions are functions of x, and the series is continued to an infinite number of terms; but the sum of all the terms after the nth may be expressed as follows, by an algebraical formula. Since /(*)=/(o)+/'(0)f+rw>s+- +/-ho)1.2/:;_1) +^(0)i^+^+1(0)o£^r) +^+i(0)rx3ST2) + -" (9) the sum of all the terms after the nth — f"(0) Z + /•» + !(()) + ... J K ) 1.2.3...n+y 1 ;1.2.3...(« + l) + = r2X^^^B+1(0)nTT + ^'<°>(. + l^ + 8) + -} (10) = 12 3* B {a quantity > /»(0) and < /»(*)} j (11) the latter factor of (10) is, I say, greater than /"(0), because the sum of the series is algebraically greater than its first term; and is less than/"(,r), because by reason of (9) /»(*) = /"(0) +/»+1(0) | +/»+»(0) + ... and with the exception of the first term of this series, each term of it is greater than the corresponding term of the series given in (10), inasmuch as the denominators are severally less. Hence representing by 6 some positive and proper fraction, we may symbolize (11) by db/"^ (12) for the latter factor is too small when 0 = 0, and is too large when 0 = 1. Hence we may write the series as follows: f{x) =/(0) +/'(0)| +/"(0)^1 + ... + _^_/-(0J.); (13) which has a definite number of terms; and therefore the only apparent difference in the absolute equality of the two sides of the equation is that which arises from 0 being an undetermined fraction greater than zero and less than unity. As this series however is of great importance in the application of the Calculus, the proof of it must not rest on any fallacious assumption, or any vague limitations which may be too wide or too narrow, such as those of the stated conditions. Hence arises the need of a rigorous and exact proof of it, and of one which will limit the extent of its applicability, and which will be given hereafter; and therefore the explanation of the last two Articles is to be considered only as yielding a presumption that such a series as (13) is likely to be true. 59.] Examples of Maclaurin's Theorem. Ex. 1. Let f(x) = (« + *)• .-. fiO) = a", fix) = nia+x)n~\ f'(0) = »«»-', fix) = n (n-l) (a+x)—», /"(0) = n(n-l)a»-a, whence, if n is positive and integral, f"ix) = n in -1) in - 2)... 3.2.1, /"(0) = n (» -1) (« - 2)... 3.2.1; fl (fl \ \ .-. fix) = ia + x)n = an + nan~1x -\ iy-^—-a"-2a?a + ... + naxn~l+ xn; (14) the common Binomial Theorem, which is therefore a particular case of Maclaurin's Theorem. Ex. 2. /(*) = e', .-. fiO) = 1, fix) = c* /'(0) = 1, fix) = e*, fidx) = e°*; • • «* = 1 +1 + Y2 + - + 1.2.3...(n-l) + r2X^e" (15> |