Hence it is clear, if n is infinite, that we must take 1 for the generating function, and that the sum of the terms of the series will differ insensibly from 1. Ex. 2. To find the sum of n terms of the series S: = Ex. 3.-To find the sum of n terms of the series S = 1.2 +2.3+3.4 + . . . . + n(n + 1). = ÷ 3+ Here we have S (1.2.3-0.1.2) 3 + (2.3.4 — 1.2.3) 3+ (3.4.5-2.3.4) ÷ 3 + . . . . + [n(n + 1)(n+ n(n + 1)(n+2) 2) — (n − 1)n(n + 1)] ÷ 3 = 3 3. We will now give the substance of the preceding remark in a more general form. To the end in view, we shall resume the series S 1 = + 1.2 and we shall represent the sum of n + 1 terms of the same series by S' 1 1 = + + 1.2 2.3 ; then, by subtraction, we get S'-S= Because the first and second members of (1) are of like 1 n + 2' when n is changed into n + 1, it is clear that (1) can be n + 2' in which C, called the arbitrary constant, is independent of n. To determine C, we put 0 for n; then, since n = 0 gives S=0, the equation S = C 1 n+1 is reduced to 0 C-1 or C= 1; consequently, putting 1 for C, we get S=1 1 n + 1 which is the same expression for the sum of n terms that we have previously found. 1 1 are the dif 1 ferences of S and which result from changing n into n + 1 n+ 1; consequently, (1) is equivalent to the equation ▲S= n + I' (2). If we denote the reverse of the method of taking the dif ference of a quantity by writing before or to the left of the quantity, then, from (2), we shall get fas=f ; because the operation indicated by A is destroyed by the reverse operation denoted byf, and that C, called the arbitrary constant, is added to the right member of the equation to correct the result, or for is the same as generality, since the difference of C It is evident that this process which is called integration is substantially the same as that previously given (though it is, perhaps, less simple), and that C can be found in the same way as before. It clearly follows from what we have done, that any term of a series is the difference of the sum of all the preceding terms taken with their proper signs, and that in order to find the sum of the preceding terms from the difference, the difference must be put in the form of an exact difference, after the manner of the right member of (1) or (2); where it may be noticed that the finding the sum of the terms from the difference is called the finding the integral, or integration. To illustrate what has been done, take the following EXAMPLES. Ex. 1.-To find the sum of n terms of the series S = (a + b) + (a + 2b) + (a + 3b) + . . . . + (a + (n − 1)6). Here we have a + nb for the difference of the sum of n terms of the series; and since (n + 1)2 — n2 = 2n + 1, we (n + 1)2 — n2 — 1 Δη-1 ; consequently, put have n = 2 = 2 ting this value for n, we have a + nb = a + 2ab+b^n2; and since n + 1 − n = 1 = ▲n, the expres 2 sion is reduced to (2a —b) An+b^n2 taking the integral, we shall get (a + nb) = C + (2a - b)n + bn2 2 or the sum of n terms of the series, is expressed by S=C+ 2a + (n-1)b 2 xn. To determine C, we put n = 0, which gives S = 0, and thence we have 0 C; consequently, the sum of n terms Ex. 2. To find the sum of n terms of the series S = 1 + 4 + 9 + 16 + 25 + . . . . + n2. Here we have AS = (n+1)2 = n2 + 2n + 1 (n + 1)3 — n3 = ▲ n3, 6n + 3 = 3(n + 1)2 — 3n2 = 3 ▲ n2, and Ans 3 An2 + An 6 (n + 1) − n = ▲n) we shall have ▲S= + 2 An3 +3 An2 + An 6 gral, we get S = 3 ; consequently, by taking the inte2n3 + 3n2 + n n(n + 1)(2n+1), noticing that it is not necessary to use any constant, since it will equal 0. SECTION X. POWERS AND INVOLUTION. (1.) ACCORDING to 9, of Sec. I., the products obtained by the successive multiplications of any number or quantity (called the root) by itself, are called powers of the root. Thus, if A is taken for the root, it is said to be the first power of A; Ax A = A2 is called the second power or square of A; A × A× A = A2 × A = As is called the third power or cube of A; A × A × A × A = A3 × A = A1 is called the fourth power of A; and generally, A × A × A x.... x A to n factors =A" is called the n'h power of A, and n, which denotes the number of times that A enters as a factor into A", is called the index or exponent of the power. (2.) The process used in obtaining any power of the root is called Involution; and when the power is formed, the root is said to have been involved or raised to the power. (3.) Because any power is formed or derived from the root by multiplication, it is evident that we shall have the following rule for raising any root to any proposed power. RULE. 1. By the rules of Multiplication, multiply the root by itself, and the product by the root again, and so on, until the number of multiplications is one less than the number of units in the index of the power; then the product thus obtained will be the required power. 2. By the rule of signs in multiplication, when the root is positive, all its powers will be positive; but if the root is negative, all the even powers, or those whose index is an even number, will be positive, and all the uneven powers, or those whose index is an uneven number, will be negative, or will have the same sign as the root. 3. If the root is a fraction, it follows from the rule for the multiplication of fractions, that the power will be found by raising the numerator and denominator to the proposed power; also, if the root is of a mixed form, or partly integral and partly fractional, it may be reduced to the form of an improper fraction, and then the power may be found as in the case of a fraction. 4. If the root consists of factors, the power may be found by raising the factors to the power, and taking their product for the sought power. 5. Because powers of the same root may be multiplied by adding their indices for the index of their product, it is clear if the root has no index expressed, that by giving it or its factors the index of the power for an index, the power will be expressed as required; also, if the root or its factors have indices, then the index of the root or the indices of its factors, when multiplied by the index of the power, will express the power, as required. 6. If we take different powers of the root, such that the sum of their indices equals the index of the power to which the root is to be raised, then the product of the power's will equal the required power of the root. Remark. It is clear from 2 of the rule that it is impossible for an even power of a number or quantity to equal a negative number or quantity; thus, a2 = — 2, x* = — a*, yo — —b⋆ are impossible equations, since there are no real values of x and y which can satisfy them; consequently, x and y are said to be impossible or imaginary. To illustrate the rule, take the following EXAMPLES. 1. To find the second, third, fourth, and fifth powers of a and - b. = Here, a xa a2, a × a × a = a2 × a = a3, a ×a×a× a = a2 × a2 = a3 × a = a1, a ×a×a×a×a = a3 × a2 = a3 are the required powers of a; and bx-b= b2, — b × — x b x − b = b2 x − b = — b3, — b x − b x − b x − b = b2 x b2 = — b3 x − b = b2, × — - -b x − b x − b x − b x − b = b2 = — b5, are the required powers of 2. To find the square of 3ab3 and the cube of Tabe |