556 A L G E B K A [THEOREMS OP EXPANSION. 1 . *i . o 1 + 2x + 3# 2 + 4< 3 + &c. *) The coefficient required is therefore that of x 5 in the last factor, viz. 6. Ex. 3. Findthesumof 1 + 2^ + 3 n -^l + 4 n(n ~ 1)(?l - 2 - } X 25 + &C. By writing 1 + 1 for 2, 1 + 2 for 3, &c., this series may be broken up into the sum of 1 + n + - - + &c. and 1 . 2 ( l (11 _ l Y 9^ 1 The latter is n { 1 + - - + ( * A " - + &c. ( l I . ^ J /. the sum required is 2 n + n1 n ~ l . 1 1 n(n 1) jE 1 ^ . 4. Find the sum of 1 + - n + - - + &c. Multiply by n + 1 ; the product is _ .*. the sum required is Ex. 5. If x r denote the product x(x - l)(x - 2) . . . (x - r + 1 ) whatever be r, and a similar notation be applied to y, and (x + y}, then , r(r-n (x + y) r = x r + rx r _ l y l + l 2 x r _ 2 y 2 + &c. We have ( 1 + a)* = 1 + a^a + -^- a 2 + r-J-x a 3 + &c. L 9 A l.^.O (1 + a)* 4z/ = their product. But (1 + a.y +y = 1 + (x + y) t a Equating coefficients of a" in the two expressions for (l + a)* +y , and multiplying by 1 . 2 . . . n, the required result is obtained. Ex. 6. If x and n be less than 1, then (1 + #)"< (1 + nx). For (1 +x) B = 1 + nx + x 2 + . . . . L 25 n(l-n) 9 A 2-n r- I 1 x I - &c.
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1.2 r. 7. On the same hypothesis (1 + #)"<- - 1 1 nx Prove that ( 1 + x)~* > - nx exactly as in the last example. Ex. 8. If x < I ; n>r< r+1; then (1 + x) H > the sum of the first r + 1 terms of the expansion ; and < the sum of the first r + 2 terms. Ex. 9. The difference between the sums of the squares of the even coefficients of the expansion of (1 +x) n , when n is an even whole number, and the sum of the squares of the odd coefficients is ( - 1 )- 2. Logarithmic Theorem. 123. The definition of a logarithm is precisely the same as that of an index or exponent (Art. 21) viz. the logarithm of a product is equal to the sum of the logarithms of the factors. Such being the case, we are at liberty to employ the definition, either in the form first given, or in the algebraic form a* =- y. In this last form x is called the logarithm of y to the index or base a. The base of the common or tabular logarithms is 10. 12-k Before proceeding to the demonstration of the theorem by which a logarithm is expressed in the form of a series, it may be as well to illustrate the definition as applied to common logarithms. 1st, Since 1 is the logarithm of 10, we may inqunv of what is -^ the logarithm; if we resume the form I0* = y } and write for x, we have to inquire what is y. Since But x 10^= y= (del) = 10, = 3. 1622777, so that the number of vhich -J is the logarithm is not a whole number, but a fraction lying between 3 and 3. In the same way, we may, but with great labour, ascer tain the numbers of which any given fraction is the logarithm. 2d, The definition will evidently enable us to obtain- a large number of logarithms, when a few have become known. For example: Given log 2 = .30 103 to find log 4 and log 5. Log 4 = log (2x2) = log 2 + log 2 (def.) = 2 log 2 = . 60206; Log 5 = log = log 10 - log 2 = l-log 2 = .69897. If in addition to log 2, log 3 be known, we can find a vast number of others. For example : Given- Log 3 = . 47712 to find log 6 and log 72 . Log 6=log 2x3 = log 2 + log 3 = . 77815, Log 72 = log 8 x 9 = 3 log 2 + 2 log 3 = 1 . 85733 . 125. To expand log (1 +x) in terms of x. Since log 1 = 0; the expansion must commence with the first power of x, the coefficient of which will depend oil the radix or base. This coefficient we shall determine afterwards for the common logarithms. In the meantime we shall denote it by A. Let then log ( 1 + x) = Ax + B^ 2 + Ca; 3 + &c. Put y + z for x ; then I. Log (1 + y + z) A(?/ + z) f 2 +Cy 3 + &c. II. Log (I I. + &c. +&c. = log (l + w) 1+2A l + y &c. + Az (1 - y + y 2 - + &c. Equating coefficients of z, y*, y~: pansions, there results A = A, 2B= -A, 31! in the two ex = -A 126. Cor. If x a - 1, where a is the base of the system, A-e have 1 =A<f a - 1 - (a- l) 2 + &c. I
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This expansion of log (l+#) is not convergent, i.e., the terms do not diminish as we advance, but the contrary, when x is any whole number greater than 1. We can, how ever, readily obtain from it a converging series for the dif
ference between the logarithms of the consecutive numbers.
