14. Find the cube root of 29 to four places of decimals, and the 16. If x be greater than y, prove that XP yp lies between px-1(x − y) and pyp −1 (x − y). to 6 places of decimals by the Binomial Theorem. 168. Before giving the proof of the Binomial Theorem when the index is fractional, it will be necessary to discuss the following preliminary proposition. be two series, in which A., A1, and Bo, B1, &c., are constant, and x a variable, which may receive any value whatever, and if these two A, = Br. + (Ar – Br) x = 0. ... Since may have any value, we may give x a value so small that (A1 − B1) x + (A2 − В¿) x2 +, &c., shall be less than any assignable quantity; hence also (A. - B.) must be less than any assignable quantity, Where Ρ and q are both positive integers, and b1, b2, &c., are the values which a1, a2, &c., take when (2-1) is written for P. = 1 + a1 (y + z) + a2 (y + z)2 + . . + ar (y + z)†, &c. = 1+z (a,+2a.y + 3a.y2+ ... + ra, yr - 1) and z. (1) y = (1 + y) —2 + a1z (1 + y) 2 q P q ૧ ... - 1 } '+az2 = (1 + y) ↑ + α ̧z (1 + b1y + b1y2+ +higher powers of y and z. +, &c. + bry"..) (2) 1+qax. ., &c. If, now, we equate the coefficients of yz, y2z, y3z, &c., in (1) and (2) agreeably to the principles of Art. (168), we shall have which is the Binomial Theorem for any fractional index, positive or negative. Examples worked out. (1 - 2x)2 1. Find the coefficient of x in the expansion of /1-3x It is evident that to obtain the coefficient of x, in the product of these two expansions, we must select the coefficients of xr-2, xr −1, x" from the second expansion, and multiply these into the numbers 4, and 1 respectively. These products, when written in their simplest form, will be 4, 2. Approximate, by the Binomial Theorem, to the value of 3/61. Now, 361 (61)3 = (64 − 3)* = {43 (1 − 3)} = に 10 (品) -243-(4)-, &c. 3 (64)3 3 644 To find the actual value of the series in decimals, we may proceed (p − q) + (p2 - q3) h + } (p2 — q3) h2 — § (p2 − q2) h2 +higher powers of h.} 1 + (p + q) h + } (p2 + pq + q2 − p − q) h2. Now, since h is very small, h2 is very much smaller than h, and all the terms after h2 are still much smaller; therefore, unless p or q be very large, 1+ (p + q) h is nearly equal to px" ql ; also, 1+ (p + q) h is nearly equal to the expansion of (1 + h)2+¶ or x2+9. CHAPTER XVII. THE MULTINOMIAL THEOREM. 170. The "Multinomial Theorem" is a formula for expanding the power of an expression which consists of any number of terms more than two, and it may be regarded as an extension of the Binomial Theorem. Before proceeding to the general investigation, we will discuss a particular case. Consider the expression (a+b+c+d+e)®. The expression of this might be obtained by multiplying together the six multinomial factors written below: (a+b+c+d+e) x (a+b+c+d+e) × (a+b+c+d+e) x (a+b+c+d+e). Now, every term in the product of these is composed of six letters, one being taken out of each line; thus there will be a number of terms a3d, each containing two a's, three b's, and one d. Now the number of terms a2b3d will be the same as the number of different ways in which we can select two a's, three b's, and one d out of the six lines above. And this number of ways is the same as the number of different orders of the letters aabbbd, taken all together; for every such order represents the formation of a term by multiplication. That is, the whole number of terms involving ab3d will be the number of permutations of six things taken all together, when two are of one sort, all alike, three of another sort, all alike, and one of another sort. |