BINOMIAL THEOREM

A formula discovered by Newton, of singular use in algebra, by which a binomial quantity may be raised to any power m, the exponent m being either a whole number or a fraction, positive or negative. The formula is this: — (a + b) z=a +— a b+ — I: '-a m (?n—l) (tm — 2) m-3,,. m(m — l) {m-~2) (to— 3) • a O'* + — 12 3 12 3 4 a"*-^b^ + . And the mere inspection of the terms will give a better idea of tlie manner in which they are succe ively formed than any explanation. When the exponent m is a whole positive number, it is evident that the series has a finite number of terms; suppose, for example, m zz 3, then »i — 3 =: 0; and all the terms into which m — 3 enters as a co-efficient become equal to zero, or vanish. But on looking at the series, we find m — 3 entering the fifth term, and it will continue in all the succeeding; therefore, when tw = 3 the series can only have four terms; and, generally, the number of terms of the series exceeds the exponent by one. When »» is a fractional number, or is negative, the series does not terminate, and will only expre approximately the value of (a + 6)"* when it is convergent; that is to say, when every individual term of the series is greater than that which succeeds it. For example, let the expre ion to be developed be (xx + t/)\ which signifies the square root of the binomial (x^ + y). Comparing this with the above formula, we have evidently — (XX + y)}t = x^i + ^ xxi-iy a series which goes on for ever; but which, supposing x greater than unity, approaches nearer and nearer to the true value of the root of x^ -|- y, as the number of terms included becomes greater. In order to determine whether in any given case the series is convergent, we have only to compare two succe ive terms of the development of (a + 6)'". For example, take the fourth and fifth; rejecting the common factors, the fourth is to the fifth as a"*"': -—^aT-^b', m — 3 b that is, as 1 to — ^ •-; therefore the fifth will be smaller than the fourth, or the series will be convergent if (m~3) b is smaller than 4 a. In general, let n be the order of any term in the development of (« + &)'": this term will be to the succeeding in the ratio of 1:: —; and n a the terms will always go on decreasing, or the series will be convergent when (/n — w + 1 ) 6 is smaller than n a. The principal use of the binomial theorem is to find approximate values of the roots of quantities by expanding them into series. The demonstrations which have been given of it are very numerous, and one or other of them may be found in any work on algebra. For one of the neatest and most concise we refer to the article " Algebra " in the Encyclopedia Britannica, 7th edition. [s. 160]