in which, P, Q, R, &c., are independent of z, and depend upon It is required to find such values 1 and m for their values. for them as will make the assumed development true for every possible value of z. If, in equation (1) we make z = 0, we have P = 1. Substituting this value for P, equation (1) becomes, (1+2)=1+ Qz+ Rz2 + Sz3 + &c. (2). Equation (2) being true for all values of z, let us make z = whence, (3). y; (1 + y)m = 1 + Qy + Ry2 + Sy3 + &c. Subtracting equation (3) from (2), member from member, and dividing the first member by (1 + z) − (1 + y), and the second member by its equal zy, we have, If, now, we make 1+z 1+y, whence z = y, the first member of equation (4), from previous principles, becomes m (1+2)-1, and the quotients in the second member become respectively, Substituting these results in equation (4) we have, (5).. m (1 + z)m-1=Q+2Rz+3S22 + 4T23+ &c. Multiplying both members of equation (5) by (1 + z), we find, m (1 + z)m = Q + 2R | z + 3S | ≈2 + 4T | z3 + &c. + 2 +2R +38 If we multiply both members of equation (2) by m, m (1 + z)m = m + m Qz + mRz2 + mSz3 + mTz1 + &c. (6). we have (7). The second members of equations (6) and (7) are equal to each other, since the first members are the same; hence, we have the equation, m+mQz+-mRz2+mSz3+&c.=Q+2Rz+3S|z2+4T|z3+ &c - (8) ÷ Q +2R] +3.S] This equation being identical, we have, (Art. 195), Substituting these values in equation (2), we 3 4 and then mul (x+a)m=x+maxm-1+ m(m-1) a2xm-2 + 1. 2 m(m—1) (m—2) + &c... (10). Hence, we conclude, since this formula is identical with that deduced in Art. 136, that the form of the development of (z+a)" will be the same, whether m is positive or negative, entire or fractional. It is plain that the number of terms of the development, when m is either fractional or negative, will be infinite. The fifth term, within the parenthesis, can be found by mul 205. The formula just deduced may be used to find an approximate root of a number. Let it be required to find, by means of it, the cube root of 31. The greatest perfect cube in 31 is 27. Let x = 27 and a = 4: making these substitutions in the formula, and putting 3 in the place of n, it becomes 31 = 3+ + + &c. or, by reducing, 3/8 27 2187 531441 43046721 Whence, 3/31: 31 = 3. 14138, which, as we shall show presently, is exact to within less than .00001. We may, in like manner, treat all similar cases: hence, for extracting any root, approximatively, by the binomial formula, we have the following RULE. Find the perfect power of the degree indicated, which is nearest to the given number, and place this in the formula for x. Sub tract this power from the given number, and substitute this difference, which will often be negative, in the formula for a. Perform the operations indicated, and the result will be the required root. 206. When the terms of a series go on decreasing in value, the series is called a decreasing series; and when they go on increasing in value, it is called an increasing series. A converging series is one in which the greater the number of terms taken, the nearer will their sum approximate to a fixed value, which is the true sum of the series. When the terms of a decreasing and converging series are alternately positive and negative, as in the first example above, we can determine the degree of approximation when we take the sum of a limited number of terms for the true sum of the series. ·d For, let ab+c-d+e-ƒ+ . . ., &c., be a decreasing series, b, c, d, . . . being positive quantities, and let a denote the true sum of this series. Then, if n denote the number of terms taken, the value of x will be found between the sums of n and n + 1 terms. For, take any two consecutive sums, c− b + c d + e −f, and abc-de-f+g. -f, are In the first, the terms which follow +gh, + k − l + . . ; but, since the series is decreasing, the terms g-h, k-l.. &c., are positive; therefore, in order to obtain the complete value of x, a positive number must be added to the sum a − b + c − d +e-f. Hence, we have a − b + c d + e -f <x. In the second sum, the terms which follow +g, are l+m. - &c., are negative; therefore, in order to obtain the sum of the series, a negative quantity must be added to abc-de-f+g; or, in other words, it is necessary to diminish it. Consequently, Therefore, x is comprehended between the sums of the first n and the first n+1 terms. x But the difference between these two sums is equal to g; and since is comprised between them, g must be greater than the difference between x and either of them; hence, the error committed by taking the sum of n terms, a b+c d+e-f of the series, for the sum of the series is numerically less than the following term. 207. The binomial formula serves also to develop algebraic expressions into series. 1 (1 − 2)−1 — 1 — 1 . (− 2) — 1 . — 1, — 1 . (− 2)2 = 1 . — 1 — 1. — 1 — 2 . (− e)3 — . . . 2 3 . or, performing the operations indicated, we find for the de velopment, 1 — — — = (1 − 2 ) →1 = 1 + 2 + 22 + z3 + 2a + &c. We might have obtained this result, by applying the rule for division. |