SECTION IX. OF SERIES. 85. From the nature of powers, (art. 14,) we readily discover that any two powers of the same root, multiplied together, produce a power indicated by the sum of the exponents of the factors; thus, a3.a3=a5; am×a"= am+n. 1 denoted by am-n; a must be equivalent to Considering the consecutive powers of a given root a as the terms of a geometrical series, extending at pleasure, above and below unity. Or the equivalent series, a-3, a-2, a-1, ao, a1, ao, a3, &c. The index denotes the order of the term beginning with unity, and making the index positive or negative, according as the quantity is above or below the point of commencement. 86. Between any two of these terms, let-n-1 mean proportionals be interposed, ex. gr. between ao (=1,) and a, calling the first x, then (art. 71,) 1: a : : 1a : x" : : 1 :x". Our series then becomes, (xo being =1=α.) X-2n x-n-1, x-n x−3, x-1, xo, x1, x2; ... ...X", x2+1, x2+3, x3n, x2n+1, &c. Now, the product of any two terms of this series is, manifestly, indicated by x, with an exponent equal to the sum of the exponents of the factors, as x1×x2+1=x3n+1; If for x and its powers we write an and its powers, the series, though changed as to the form of its exponents, evidently retains the same essential character, and the operations of multiplication and division are performed by taking the sum and difference of the exponents of the factors, whether those exponents are positive or negative, integral or fractional. By thus arranging the powers as the terms of a geometrical series, we perceive that the exponent, whether integral or fractional, serves not only to designate the power, but to indicate the situation of the term, in relation to the unit's place. 87. From these principles we readily infer, that any letter or quantity may be removed from the denominator of a fraction to its numerator, and vice versa, by changing the sign of its exponent. 88. It frequently happens in the division or evolution of algebraic quantities, as well as in common arithmetic, that to whatever extent the process may be continued, a remainder will still occur; in which case the resulting quotient or root, mostly assumes the form of an infinite series. 1- X -=1+x+x2+x3+x+, &c. to infinity. b3 b4 be 568 ✔a3+b2=a+ + 2a 8a3 16a5 128a7 +, &c.* In some instances, when a few terms of the series are obtained, the law of continuation, or the relation of the successive terms to those which precede them, becomes manifest. The first of the series above given, may be readily continued to any proposed extent. The law of continuation in the last is not obvious on first view; it will, however, be shown further on. 89. In the investigations connected with series, the method of indeterminate co-efficients is often found particularly convenient. It depends upon the following theorem. Let Ax+Bx2+Сx3+Dx2, &c. =ax+bx2+cx3+dx2, &c., the series being both infinite, or, if finite, extending to the same number of terms; and A, B, C, a, b, c, invariable; if then the above equation be true, whatever value may be assigned to x, A will be equal to a, B⇒b, &c.; for dividing by x, we have A+Bx+Cx2+Dx3, &c.=a+bx+cx2+dx3, &c. Now, the equation being true for all values of x, must hold if x=0, in which case, the equation becomes A=a; subtracting this equation from the given one, Bx+Cx2+Dx3, &c.=ax+bx2+cx3, &c. * When the successive terms of an infinite series continually decrease, it is called a converging series; in which case, the sum of the series may be approximated by collecting a finite number of terms. Dividing by x, B+Сx+Dx2; &c.=b+cx+dx2, &c. Whence, if x=0, B=b. In the same manner C=c, D=d, &c. This equation becomes by transposition, A x + Bx2 + Ca2 + Dx++; &c.}=0. S Hence it appears, that when all the terms of a general equation are brought to one side, the co-efficients of the several powers of the unknown quantity are respectively equal to 0. This principle is applied in the following examples. 1 1. Required to express 1=2x+x2 in a series. It is easy to perceive that the first term must be 1. 1 Assume then =1+ax+bx2+cx3+dx1, &c. -2x+x2 Multiply by 1-2x+x2, and bring the terms to one side of the equation, Then, 1+ax+bx2+cx2+dx2+, &c. -1 -2x-2ax2-2bx3—2cx1—, &c. }=0. Hence, a-2=0,b-2a+1=0, c-2b+a=0,d-2c+b=0. .. a=2, b=4—1=3, c—6—2—4, d—8—3—5. Whence, 1 1-2x+x3 =1+2x+3x2+4x3+5x++, &c. in which the law of continuation is manifest. 3. Required to develope ✅a2+x2, in a series. Assume 1+z2=1+az2+bza+cz®+dz®+, &c.* By involution and transposition, 1+2az2+2bz1+2cz +2dz3+2ez10+, &c.) .. 2a-1=0,2b+a2=0,2c+2ab=0,2d+2ac+b2=0,&c. a+ + 5x8 7x10 &c. 2a 8a3 16a3 128a7 256a99 To find, if possible, the law of continuation, we observe that * If the series 1+az+bz3+cz3, &c. had been assumed, we should have found a=0, c=0, &c. |