SUMMATION OF INFINITE SERIES. The summation of series of this kind is the finding of a finite expression equal to the proposed series, and in many cases this finite expression is found by subtraction. 3 4 5 6 Hence, by subtracting the latter from the former, we have 1 (2.) Required the sum of the series 1 + 1 1 + = +1'n+2 3 + 5 ... 2 + ( + n 4n+8 n+2 (1+ 1 1 1 -- + + 1.3 2.4 3.5 4.6 (7.) Sum the series 1.+8z+5x2+7c23+9x1 .... ad infinitum. INVESTIGATION OF THE BINOMIAL THEOREM. 194. Let it be required to expand (a+x)" in a series, whether n be integ‐ ral or fractional, positive or negative. Since (a+x)" = { a(1+¤) } "=a"(1+2)", whatever n may be, if (1+y)=1+Ay+By2+Cy3+Dy1+ then (a+x)"a" · (1+5)* ... Case 1. Let n be integral and positive. Then (1+y)=(1+y).(1+y).(1+y) to n factors; and by effecting the multiplication of a few of these equal factors, we shall find that the first two terms of the series are 1+ny, and that the remaining terms are of the form By2+Cy3+Dy1+ .... where B, C, D, .... are undeter mined coefficients, entirely independent of y; and therefore we have A=”. Case 2. Let n be integral and negative. (i+y)”=(1+y)}=1+Ay+By2+Cy3+....... .. (1+y)=(1+Ay+By2+Cy3+....) a =1+q(A+By+Cy2+....)y+, &c. Hence, equating the coefficients of like powers of y, we have Hence A=-=n, and therefore in all cases A=n; and, consequently, 9 (a+z)"=a"+naa—1x+Ba"—2x2+€c®-3x3 + · · .... Now in order to determine the values of the coefficients B, C, D, &c., we have (a+x+z)"={(a+x)+z}"={a+(x+2)}", and if we expand according to each of these forms, the two expansions must be identical; hence, by the first form we have (a+x+z)"={(a+x)+z}" = (a+x)"+na+x)1z+B(a+x)^_2 z2+C(a+x)¤—31⁄23+ ..... +B{a+(n-2)a"3x+......}z2 + &c. =a"+na"1x+Ba"—22:2 +Ca33 +Da”—1x1+ +naz+n(n-1)axz+Ba3x2z+ +B(n-2)a-3x22+ ... ... Again: (a+x+z)"={a+(x+z)}" +Can-323 + .... &c. =a"+na"1(x+2)+Baa2(x+2)2+Caa3(x+z)3+ .... =u"+na"-1 x+ Baa—2x2+ Caa3μ3+ . and the coefficients of the same powers of x and z, in these two expansions, must be the same (Art. 191); hence we have n (n-1) n(n-1) (n-2) 1. 2. 3 n (n-1) (n-2) (n-3) 1. 2. 3. 4 &c. 4 D= (n−3) C .. D = which is the Binomial Theorem, and where the last term represents the (p+1)th term of the expansion. Hence (a—x)" =a”—na"-1x+n(n—1) a"-2x2_n(n−1) (n−2) a13μ3+ &c. and in all these formulæ n may be either integral or fractional. THE EXPONENTIAL THEOREM. 195. It is required to expand a3 in a series ascending by the powers of 1. Since a=1+a-1; therefore a={1+(a−1)}2, and by the Binomial Theorem we have {1+(a−1)}2=I+x(a−1)+2(x − 1 ) (a−1)2 + ~(x−−1) (x—2) (a−1)3 + .... 1.2 1.2.3 =1+{(a-1)-(a−1)2 + ÷ (a−1)3—4(a−1)' + ....} x + Br2 + Cz3.. where B, C,.... denote the coefficients of x2, x3, . . . . .; and if we put Then a=1+Ax+Bx2+Cx3+Dx1+Ex2+ ..... ..... ax+h=1+A (x + h) + B (x + h)2 + C (≈ + h)3 + + Ah +2 Bxh + 3Cx2h + 4Dx3h +Bh2+3Cxh2 + 6Dxh2 .... + + + +Cha+4Dxh3 ... But ax+b=a*xa1=(1+Ax+Bx2+Cx3+ •·) (1+Ah+Bh2+Ch3+ =1+ Ax + B x2 + C 23 + Dx1 + + Ah + A2xh + ABx2h + ACx3h + +Bh2 + ABxh2 + B2x2h2 + + Ch3 +A Сxh3 + + Dh1 + .... Now these two expansions must be identical; and we must, therefore, have the coefficients of like powers of ≈ and h equal; hence which is the Exponential Theorem; where A=(a−1)—† (a − 1 )2 + ↓ (a — 1 )3 — 4 (a — 1) + Let be the value of a, which renders A=1, then (ε — 1) — ÷ (e− 1 )2 + ↓ ( e − 1 ) 3 − ‡ (e − 1 )1 + . . . =1 1.2 12:3 184 Now, since this equation is true for every value of x, let x=1; then LOGARITHMS. 196. LOGARITHMS are artificial numbers, adapted to natural numbers, in order to facilitate numerical calculations; and we shall now proceed to explain the theory of these numbers, and illustrate the principles upon which their properties depend. DEFINITION. In a system of logarithms, all numbers are considered as the powers of some one number, arbitrarily assumed, which is called the BASE of the system, and the exponent of that power of the base which is equal to any given number is called the LOGARITHM of that number. Thus, if a be the base of a system of logarithms, N any number, and a such that N = a* then x is called the logarithm of N in the system whose base is a. The base of the common system of logarithms, (called from their inventor "Briggs's Logarithms"), is the number 10. Hence since From this it appears, that in the common system the logarithms of every num ber between 1 and 10 is some number between 0 and 1, i. e. is a fraction. The logarithm of every number between 10 and 100 is some number between 1 and 2, i. e. is 1 plus a fraction. The logarithm of every number between 100 and 1000 is some number between 2 and 3, i. e. is 2 plus a fraction, and so on. 197. In the common tables the fractional part alone of the logarithm is registered and from what has been said above, the rule usually given for finding the characteristic, or, index, i. e. the integral part of the logarithm will be readily understood, viz. The index of the logarithm of any number greater than unity is equal to one less than the number of integral figures in the given number. Thus, in searching for the logarithm of such a number as 2970, we find in the tables opposite to 2970 the number 4727564; but since 2970 is a number between 1000 and 10000, its logarithm must be some number between 3 and 4, i, e. must be 3 plus a fraction; the fractional part is the number 4727564, which we have found in the tables, affixing to this the index 3, and interposing a decimal point, we have 3.4727564, the logarithm of 2970. |