PART VII. ALGEBRA. INFINITE SERIES". A 1. SERIES is a rank of quantities, which usually proceed according to some given law, increasing or decreasing successively; the simple quantities which constitute the series are called its terms. 2. An increasing or diverging series is that in which the terms successively increase, as 1, 2, 3, 4, &c. a+3a+7a, &c. 3. A decreasing or converging series is that in which the terms successively decrease, as 5, 3, 1, &c. 10 a—7 a—2 a, &c. The doctrine and application of infinite series, justly considered as the greatest improvements in analysis which modern times can boast, were introduced about the year 1668, by Nicholas Mercator, who is supposed to have taken the first hint of such a method from Dr. Wallis's Arithmetic of Infinites; but it was the genius of Newton that first gave it a body and form. The principal use of infinite series, is to approximate to the values and sums of such fractional and radical quantities, as cannot be determined by any finite expressions; to find the fluents of fluxions, and thence the length and quadrature of curves, &c. Its application to astronomy and physics is very extensive, and has supplied the means whereby the modern improvements in those sciences have been made. The intricacy of this branch of science has exercised the abilities of some of the most learned mathematicians of Europe, and its usefulness has induced many to direct their chief attention to its improvement: among those authors who have written on the subject, the following are the principal; D'Alembert, Barrow, Briggs, the Bernoullis, Lord Brouncker, Bonnycastle, Des Cartes, Clairaut, Colson, Cotes, Cramer, Condorcet, Dodson, Euler, Emerson, Fermat, Fagnanus, Goldbach, Gravesande, Gregory, Halley, De l'Hôpital, Harriot, Huddens, Huygens, Horsley, Hutton, Jones, Kepler, Keill, Kirkby, Landen, De Lagni, Leibnitz, Lorgna, Manfredi, Monmort, De Moivre, Maclaurin, Montano, Nichole, Newton, Oughtred, Riccati, Regnald, Saunderson, Slusius, Sterling, Stuart, Simpson, Taylor, Varignon, Vieta, Wallis, Waring, &c. &c. 4. A neutral series is that in which the terms neither increase nor decrease, as 1, 1, 1, 1, &c. a+a+a+a, &c. 5. An arithmetical series is that in which the terms increase or decrease by an equal difference, as 1, 3, 5, 7, &c. 9, 6, 3, 0, &c. a+2a+3 a, &c. 6. A geometrical series is that in which the terms increase by constant multiplication, or decrease by constant division, as 1, 3, 9, 27, &c. 12, 6, 3, 3 2 2 &c. a+2a+4a+8 a, &c. 7. An infinite series is that in which the terms are supposed to be continued without end; or such a series, as from the nature of the law of increase or decrease of its terms requires an infinite number of terms to express it. 8. On the contrary, a series which can be completely expressed by a finite number of terms, is called a finite or terminate series. 9. Infinite series usually arise from the division of the numerator by the denominator of such fractions as do not give a terminate quotient, or by extracting the root of a surd quantity. 10. To reduce fractions to infinite series. RULE I. Divide the numerator by the denominator, until a sufficient number of terms in the quotient be obtained to shew the law of the series. II. Having discovered the law of continuation, the series may be carried on to any length, without the necessity of farther division. 1. a-b ́=a1 — 1 + a" — 2b + an — 3b2+, &c. to + b2 · dently terminates. 2. an-bu ·2b+ a¤ — 3 b2 —, &c. which terminates in—ba— 1, when n is an even number, but goes on indefinitely when n is odd. a" + bu 3. a+b ·a1 — 2 b + aa — 3 b2 —, &c. which series terminates in −1, when n is an odd number, but goes on indefinitely when n is even. 1+x)1 (1-x+x2-x3+, &c. the series required. 4. The difference a"-b" is not measured by the sum a+b. Hence, first, the difference of the nth powers of any two numbers is measured by the difference of the numbers, whether n be even or odd. Secondly, it is measured by the sum of the numbers, when n is even, but not when n is odd. Thirdly, the sum of the nth powers is measured by the sum of the numbers when n is odd, but not when n is even. In each of the quotients which terminate, the number of terms is equal to the index n. See an ingenious application of these conclusions in the Rev. Mr. Bridge's Lectures on Algebra, p. 248. 11. When any quantity is common to every term, the series may be simplified by dividing every term by that quantity, putting the quotients under the vinculum, and placing that quantity before the vinculum, with the sign x between. ent put under the vinculum and connected with the divisor · a sign ×, the series becomes x1+ + ·+ +, &c. which is a simpler form than that in the example. 1 -to an infinite series. Ans. 1+x+x2+x3+,&c. 3. Reduce 1 Ꮖ α 5. Let be converted into an infinite series. Ans. x+z be turned into an infinite series. Ans. X 12. To reduce compound quadratic surds to infinite series. RULE. Extract the square root, (Art. 57. Part 3. Vol. I.) and continue the work until the law of the series be discovered; after which the root may be carried to any length, as in the preceding rule, and it will be the series required. EXAMPLES.-1. Convert x2+22+ to an infinite series. 2. Let a2-x2 be converted into an infinite series. Ans. a 3. Change a2+b into an infinite series. Ans. a+ 4. Express 1++ in an infinite series. 13. SIR ISAAC NEWTON'S BINOMIAL THEOREM *. For readily finding the powers and roots of binomial quantities. RULE I. Let P=the first term of any given binomial, Q= the quotient arising from the second term being divided • This theorem was first discovered by Sir I. Newton in 1669, and sent (in the above form) in a letter dated June 13th, 1676, to Mr. Oldenburgh, at that time Secretary of the Royal Society, in order that it might be communicated to M. Leibnitz. As early as the beginning of the 16th century, Stifelius and others knew how to determine the integral powers of a binomial, not merely by continued multiplication of the root, but also by means of a table, which Stifelius had formed by addition, wherein were arranged the coefficients of the terms of any power within the limits of the table. Vieta seems also to have |