502. In Chap. xxxix, we investigated the Binomial Theorem when the index was any positive integer; we shall now consider whether the formulæ there obtained hold in the case of negative and fractional values of the index: Since, by Art. 385, every binomial may be reduced to one common type, it will be sufficient to confine our attention to binomials of the form (1+x)". By actual evolution, we have 1 x2 + 1 -osovce; 16 and by actual division, 1 (1 — x)2 and in each of these series the number of terms is unlimited. In these cases we have by independent processes obtained an expansion for each of the expressions (1+x)} and (1+x)? We shall presently prove that they are only particular cases of the general formula for the expansion of (1+x)”, where n is any rational quantity. This formula was discovered by Newton. 503. Suppose we have two expressions arranged in ascending powers of x, such as m m(m-1)(m-2) 23+ 1+mx + ......(1), 1.2 1.2.3 n(n-1) (n - 2) and 1+ nx + (2) 1.2 1.2.3 n(n − 1) x2+ -23+ The product of these two expressions will be a series in ascending powers of denote it by 1+Ax+ Bx2+ Cx3+ Dx4 +-co...; then it is clear that A, B, C, ...... are functions of m and n, and therefore the actual values of A, B, C, ...... in any particular case will depend upon the values of m and n in that case. But the way in which the coefficients of the powers of x in (1) and (2) combine to give A, B, C, ...... is quite independent of m and n; in other words, whatever values m and n may have, A, B, C, - preserve the same invariable form. If therefore we can determine the form of A, B, C, ...... for any value of m and n, we conclude that A, B, C, ...... will have the same form for all values of m and n. The principle here explained is often referred to as an example of “the permanence of equivalent forms; ” in the present case we have only to recognize the fact that in any algebraical product the form of the result will be the same whether the quantities involved are whole numbers, or fractions ; positive, or negative, We shall make use of this principle in the general proof of the Binomial Theorem for any index. The proof which we give is due to Euler. 504. To prove the Binomial Theorem when the index is a positive fraction. Whatever be the value of m, positive or negative, integral or fractional, let the symbol f(m) stand for the series mm-1) ( 1.2.3 then f(n) will stand for the series n(n-1) n(n-1)(n-2) 1+na+ -X2 + 1.2 1.2.3 If we multiply these two series together the product will be another series in ascending powers of X, whose coefficients will be unaltered in form whatever m and n may be. To determine this invariable form of the product we may give to m and n any values that are most convenient; for this purpose suppose that m and n are positive integers. In this case f(m) is the expanded form of (1+x), and f(n) is the expanded form of (1+x)"; and therefore -22 + f(m) x f(n)=(1+x)" X (1+x)"=(1+x)m+n, but when m and n are positive integers the expansion of (m+n)(m+n–1)x2 + (1+x)+n is 1+ (m+n+ 1.2 This then is the form of the product of f(m) x f(n) in all cases, whatever the values of m and n may be; and in agreement with our previous notation it may be denoted by f(m+n); therefore for all values of m and n f(m) xf(n)=f(m+n). Also fm) xf(n) (P)=f(m+n) xf(p) =f(m+n+p), similarly. Proceeding in this way we may show that f(m) x f(n) xf(p)... to k factors = f(m+n+p+ ... to k terms). h Let: each of these quantities m, n, p, be equal to where E' h and k are positive integers; {*(*) }*=f(h); but since h is a positive integer, f(h)=(1+x)"; k ; .. (1+x)={s(7)} .. (1+2)&=r(); but f (*) stands for the series - x2 + ......; nh h k Ik 1.2 1 1.2 which proves the Binomial Theorem for any positive fractional index. :: (1+x)=1+ 505. To prove the Binomial Theorem when the index is any negative quantity. --n It has been proved that f(m) xf(n)=f(m+n) for all values of m and n. Replacing m by (where n is positive), we have f(-n) xf(n)=f(-n+n) =f(0) =1, since all terms of the series except the first vanish; 1 (; f(n but f(n)=(1+x)", for any positive value of n; 1 =f(-n), (1+x)" (1+x)-n=f(-n). But f(-n) stands for the series 1+(-n)x+ (-n)(-n-1) ...; 1.2 22+ 1.2 which proves the Binomial Theorem for any negative index. Hence the theorem is completely established. or -22+ ; CHAPTER XLVIII. EXPONENTIAL AND LOGARITHMIC SERIES. 506. The advantages of common logarithms have been explained in Art. 410, and in practice no other system is used. But in the first place these logarithms are calculated to another base and then transformed to the base 10. In the present chapter we shall prove certain formulæ known as the Exponential and Logarithmic Series, and give a brief explanation of the way in which they are used in constructing a table of logarithms. 507. To expand ain ascending powers of x. By the Binomial Theorem, if n>1, 1 na + n hence the series (1) is the wth power of the series (2); that is, 2 |