which establishes the Binomial Theorem in its most general form. REMARK. From the preceding formula and demonstrations, corollaries, similar to those in Art. 310, may be drawn, but it is not necessary to repeat them. The following additional proposition is sometimes useful. ART. 320. To find the greatest term in the expansion of (a+b)". From Cor. 3, Art. 310, we have seen that the rth term is n(n-1). (r-1) from the general law, the (r+1)th term is n(n-1).. (n—r+1)an-rbr. r Therefore, the (r+1)th term is derived from the rth by multiplying 1= a statement, or the latter by nr+1 b While this multiplier is greater than 1, each term must be greater than the preceding. Hence, the 7th term will be the great Take r, therefore, the first whole number greater than b (n+1) and the 7th term will be the greatest of the series. a+b2 If (n+1). is a whole number, then two terms are equal, b each of which is greater than any of the other terms. Ex. 1. Find the greatest term in the expansion of (1+1)1⁄2o. Ans, 2nd 10 2. Find the greatest term in the expansion of (1+%) 3. Find the greatest term in the expansion of (3+5x)3, when x=1. Ans. 5th Cor. 1. By finding the greatest term of a series, we determine the point at which the series begins to converge; that is, the point from which the terms become less and less. Cor. 2. It is also evident that when n-r+1 is first less than 1, that the coefficient of the preceding term is the greatest. +1=213, denotes the term having the greatest than 201+1=258 coëfficient. If n is an odd integer, there will be two coefficients, the n+1th and the n+3a, each greater than any other. 2 Ex. Find the term having the greatest coëfficient in the expansion of (a+b)1o; and the two terms having the greatest coëfficients in the expansion of (x-y)'. Ans. 6th, and 4th and 5th. ART. 321. In the application of the Binomial Theorem, it is merely necessary to take the general formula (a+b)"=a"+na"-1z +, &c., and substitute the given quantities instead of the symbols to which they correspond in the formula, and then reduce each term to its most simple form. Ex. 2. Develope (1-x). Here a=1, b——x, n=—1. In making these developments it will assist the pupil, to recollect that every root and every power of 1, is 1. Ex. 3. Develope Ja+b into a series. Since a+b=a(1+-), .. √a+b=√ā(1+b)' Here a=1, b=b-, n=11. x3 2x6 3a3 36a6 3 6·9a? 2215 21 1,101 2/9=2/8+1=2+ 3 8 9 82 81 83 (a3—x3)3=a(1—23, +; a3 = (a3—x3) 13 3 ART. 322. To find the approximate roots of numbers by the Binomial Theorem. Let N represent any proposed number whose nth root is required, take a such that a" is the nearest perfect nth power to N, so that N=a"±b, b being small compared with a, and+ or -, according as N> or <a"; then Na(1), by writing b an for b in the general Of this series a few terms only, when b is small with regard to a", will give the required root to a considerable degree of accuracy. Ex. Required the approximate cube root of 128. 3/128=2/5+3=53/1+,2; Here ART. 323. In the preceding example, since the series continues to infinity, we obtain only an approximate value for the required root, and as the denominators increase more rapidly than the numerators, a few terms only need be taken for practical purposes; still it may be required to find what is the limit in the error occasioned by neglecting the remaining terms of the series. To do this let R be the true root, and as the terms are alternately positive and negative, let Ra-bcd+e-f+g-h+k+, &c., and let R'a-b+c-de-f, R"-a-b+c-d+e—fƒ +g. Then since the terms continually decrease, a-b, c—d, e—f, gh, &c., are all positive, and therefore R', which contains three only of those differences, will be less than R. For the same reason all the pairs of terms after g, as ―h+k, —l+m, &c., will be all negative, and R" will be greater than R; therefore, the true value of the series lies between R' and R", or and a-b+c―d+e—f, a-b+c-d+ef+g. Hence, the error committed by the omission of any number of the terms of a converging series, is less than the first term of the omitted part of the series. Thus, in the preceding example, if we had stopped at the second term, the error would have been less than .0000042. 15. Find the 5th root of 35. Here Ans. 2.036172+. N=35=32+3=23 (1+3). 16. The student may solve the following examples :/ 3 (1). √10 =√9+1 =3.16227 true to 0.00001. 3/27-3-2.88449 (4). 4/260=4/256+4=4.01553 (5). 7/108=7/128-20-1.95204 X true to 0.00001. .. true to 0.00001. all le REMARK. Instead of extracting the nth root by the formula in Art. 322, the operation may be performed by the general formula of the preceding article, the number whose root is to be extracted being divided into any two parts whatever. The advantage of the formula in Art. 322 consists in the rapid convergence of its terms. Thus in finding the 4th root of 260 true to five places of decimals, it is only necessary to take two terms of the series. |