n n I' +1 equal in this case to; but as the above series is not converging, it cannot give this true value; and we must, therefore, take into consideration the remainder, at whatever term we stop. If we suppose, in the preceding series, n = 2, we have 1− } + { − } + 1, − &c., a series, in which the particular sums, 1, 1, 3, §, &c. are, alter nately, smaller and greater than the true value of , but to which they approach continually, because series is converging. n which is n + 1' Although diverging series, that is, those, the terms of which go on increasing, continue to depart further and further from the true value of the expressions from which they are derived, yet considered as developements of these expressions, they may serve to show such of their properties, as do not depend on their summation. 237. If we continue any process of division in algebra, according to the method pursued above (235), with respect to the quantities m and m-1, the quotient will always be expressed by an infinite series composed of simple terms. Infinite series are also formed by extracting the roots of imperfect powers, and continuing the operation upon the several, successive remainders; but they are obtained more easily by means of the formula for binomial quantities, as will be shown in the Supplement, where I shall treat of the more common series. Examples in Arithmetical Progressions. Let a denote the first, 7 the last term, n the number of terms, d the common difference, and S the sum of an arithmetical progression. n n 1 = a + (n − 1) 8, S = (a + 1) = (2a + (n − 1) d) 1⁄2. 2 1. Given a = 1, 81, n=14; then l 14, S=105. n = 17; then = 50, S=442. 5. Given a 2, 8 = 1, n = 26; then 731, S= 601. δ 6. Given a 4, 8=1, n = 13; then / 20%, = 139. S 7. Given a=-7,8=3, n = 8; then 714, S=28. 8. Given a 6, 53, n = 30; then = 15%, S 1461. · = 7= = 9. Given a 1, d=-1, n = 20; then 7=-17, = S-133. 10. Given a 31, d=-25, n = 15; then l=-361, S=-2471. 11. Given a = 0, 8 = 1, n = 11; then l=5, S = 27. 12. Given a = — 10, 8= 2, n = 6; then 7 —— 20, , n=25; then 7 —— 213, S2811. Examples in Geometrical Progression. Let a be the first term, q the common ratio, l the last term, n the number of terms, and S the sum of all the terms; then 1. Given a = 1, q = 2, n = 7; then 7 = 64, S 127. 4, q = 3, n = 10; then = 78732, Given a = S=118096. 3. Given a 9, q=%, n = 7; then 72582073, S=59174. 4. Gren a 61, q = 1, n = 8; then = 106, S = 307411. 3 5. Give a = 6, q = 2, n = 6; then 7 = 1317, S= 19373. 6. Give a 5, q= 4, n = 9; then = 327680, = S=436905. Theory of Exponential Quantities and of Logarithms. 238. In the seveal questions we have resolved thus far, the unknown quantities lave not been made subjects of consideration as exponents; this wil. be requisite, however, if we would determine the number of terms in a progression by quotients, of which the first term, the last term, and the ratio are given. In fact, we are furnished by a question of this kind with the equation - 1 = a qu―1 (231), in which n will be the unknown quantity; abridging the expression, by making n − 1 = x, we have la q x. This equation cannot be resolved by the direct methods hitherto explained; and quantities like a cannot be represented by any of the signs already employed. In order to present this subject in a more clear light, I shall go back to state, according to Euler, the connexion which exists between the several algebraic operations, and the manner, in which they give rise to new species of quantities. 239. Let a and b be two quantities, which it is required to add together; we have a + b = c; and in seeking a or b from this equation, we find hence the origin of subtraction; but when this last operation carnot be performed in the order in which it is indicated, the reult becomes negative. The repeated addition of the same quantity gives rise to multiplication; a representing the multiplier, b the multiplican', and c the product, we have and hence arises division, and fractions, in which thi division terminates, when it cannot be performed without a rexainder. The repeated multiplication of a quantity by itslf produces the powers of this quantity; if b represent the number of times a is a factor in the power under consideration, we hav abc. This equation differs essentially from the preeding, as the quantities a and b do not both enter into it of the ame form, and hence the equation cannot be resolved in the same way with respect to both. If it be required to find a, it may be obtained by simply extracting the root, and this operation give rise to a new species of quantities, denominated irrational; but I must be determined by peculiar methods, which I shall proceed to illustrate, after having explained the leading properties of the equation abc. 1 240. It is evident, that if we assign a constant value greater than unity to a, and suppose that of b to vary, as may be requisite, we may obtain successively for c all possible numbers. Making b = 0, we have c = 1; then since b increases, the corresponding values of c will exceed unity more and more, and may be rendered as great as we please. The contrary will be the case, if we suppose b negative; the equation ac being then changed into a = c, or = c, the values of c will go on decreasing, and may be rendered indefinitely small. We may, therefore, obtain from the same equation all possible positive numbers, whether entire or fractional, upon the supposition, that a exceeds unity. The same is true, if we have a < 1; only the order in which the values stand, will be reversed; but if we suppose a = 1, we shall always find c = 1, whatever value be assigned to b; we must, therefore, consider the observations which follow, as applying only to cases, in which a differs essentially from unity. In order to express more clearly, that a has a constant value, and that the two other quantities b and c are indeterminate, I shall represent them by the letters x and y; we then have the equation a2 = y, in which each value of y answers to one value of x, so that either of these quantities may be determined by means of the other. 241. This fact, that all numbers may be produced by means of the powers of one, is very interesting, not only when considered in relation to algebra, but also on account of the facility with which it enables us to abridge numerical calculations. Indeed, if we take another number y', and designate by the corresponding value of x, we shall have a = y', and consequently, if we multiply y by y', we have y y = a* x ax' = a*+*; if we divide the same, the one by the other, we find lastly, if we take the mth power of y, and the nth root, we have It follows from the first two results, that knowing the exponents x and belonging to the numbers y and y', we may, by taking their sum, find the exponent which answers to the product y y', and by taking their difference, that which answers to the quotient y. From the last two equations it is evident, that the exponent belonging to the mth power of y may be obtained by simple multiplication, and that which answers to the nth root, by simple division. y Hence it is obvious, that by means of a table, in which, against the several numbers y, are placed the corresponding values of x, y being given, we may find x, and the reverse; and the multiplication of any two numbers is reduced to simple addition, because, instead of employing these numbers in the operation, we may add the corresponding values of x, and then seeking in the table the number, to which this sum answers, we obtain the product required. The quotient of the proposed numbers may be found, in the same table, opposite the difference between the corresponding values of x, and, therefore, division is performed by means of sub traction. These two examples will be sufficient to enable us to form an idea of the utility of tables of the kind here described, which have been applied to many other purposes since the time of Napier, by whom they were invented. The values of x are termed logarithms, and, consequently, logarithms are the exponents of the powers, to which a constant number must be raised, in order that all possible numbers may be successively deduced from it. The constant number is called the base of the table or system of logarithms. I shall, in future, represent the logarithm of y by ly; we have then xly, and since y = a, we are furnished with the equation y = aly. 242. As the properties of logarithms are independent of any particular value of the number a, or of their base, we may form an infinite variety of different tables by giving to this number all possible values, except unity. Taking, for example, a = 10, we have y = (10), and we discover at once, that the numbers 1, 10, 100, 1000, 10000, 100000, &c., which are all powers of 10, have, for logarithms, the numbers 0, 1, 2, 3, 4, 5, &c. |