Also right side = A + By + Cy2+ y3 x integral expression in y. Hence, equating coefficients of yo, y1, y2, we have Ex. 1. Find the coefficient of x in the expansion of according to ascending powers of x. 1 1-5x+6x2 ... - 2 {1+ 2x + (2x)2 + + (2x)" + ...}. ... Hence the required coefficient is 3n+1-2n+1. Ex. 2. Find the coefficient of x2+r in the expansion of From Ex. 2, Art. 295, we have (1 + x)n 3n 1 = n 3n-1 1 n (n − 1) 3n-2 1 + (1 - 2x)2 + an integral expression of the (n-3)th degree. Whence the required result. Ex. 3. Shew that the sum of all the homogeneous products of n dimensions of the three letters a, b, c is equal to an+2 (c − b )+bn+2 ( a − c ) + c2+ 2 (b − a) (bc) (ca) (a - b) The sum of all the homogeneous products of n dimensions is the coefficient of x" in the product (1+ax+ a2x2 + ...) (1 + bx + b2x2 + ...) (1+cx + c2x2 + ...) [See Art. 289]; (a - b) (a - c) 1 1 + b2 - ах (b-c) (b-a) 1-bx (ca) (c-b) 1-cx and the coefficient of x" in the expansions of these partial fractions is easily seen to be Ex. 4. To find the sum of all the homogeneous products of n dimensions which can be formed from the r letters a1, a2, A3,................., Ap. As in the previous example, the sum required will be the co 297. Indeterminate coefficients. We shall conclude this Chapter by giving two examples to illustrate a method, called the method of indeterminate coefficients, which depends upon the theorems established in Articles 91 and 277. Ex. 1. Find the coefficient of x" in the expansion, according to ascending powers of x, of (1+cx) (1+c2x) (1 + c3x)... (1+ c2x). The continued product is of the nth degree in x; we may therefore assume that (1 + cx) (1+c2x).......(1+c2x) = A ̧+A ̧¤+A„x2+ ... +Anxn, Now change x into cx; then, since A。, A1, A2, &c. do not contain x, we have (1+c2x) (1 + c3x)..... (1+c2+1x) = A。+  ̧¢œ+ А„c2x2 + ..... ... = (1+cx) (A。 + A ̧¤¤ + A‚c2x2 + + Âμc2x2 + ... + Anc2x2). ... Now equate the coefficients of " on the two sides of the last identity, and we have Ex. 2. To find the sum of the series 12+22+32+ ... +n2. Let 12+22+32 + +n2 = à ̧n+A‚n2 +Á ̧n3....... (a) for some particular value of n, where A1, A2, Ag do not contain n. The relation (a) will be true for n+1 as well as for n, provided 12+22+32+...+n2 + (n+1)2= A1 (n+1)+ A ̧ (n+1)2 + A ̧ (n + 1)3; or, subtracting (a), provided 2 (n+1)2=A ̧ +(2n+1) A2+ (3n2 +3n+1) A ̧. Now the last relation will be true for all values of n if we give to A1, A2, A, the values which satisfy the equations found by equating the coefficients of n2, n1 and no, namely, the equations 34, 1, 34, +24=2, and A ̧+A2+A1=1, from which we obtain 64,=242=3A3=1. 1 1 Hence, if the relation 12+22 + ... + n2 = n+ n2+n3, be true 6 2 3 for any value of n, it will be true for the next greater value. But it is obviously true when n=1; it will therefore be true when n=2; and, being true when n=2, it must be true when n=3; and so on indefinitely. The sum of the cubes, or of any other integral powers, of the first n integers can be found in a similar manner. 17. Find the coefficient of x" in the expansion of x + 4 x +5x+6 18. Find the coefficient of x" in the expansion of 2n 19. Shew that the coefficient of x2-1 in the expansion of x+5 (x2 - 1)(x+2) of 1 is 1 20. Find the sum of the n first coefficients in the expansion 3- 2x 1- 2x-3x2 21. Find the sum of the n first coefficients in the expansion 22. Find the coefficient of x" in the expansion of Find also the sum of the n first coefficients. +r (1 + x)" (1-x)3° 23. Shew that the coefficient of x" in the expansion of n-1 25. Shew that the coefficient of 2"-1 in the expansion of 26. {(1 − ≈) (1 − cz) (1 − c2z) (1 − c3z)}" is Prove that a (b − c) (bc – aa') (aTM – a'TM) ̧ b (c− a) (ca – bb′) (bTM — b’m) = a - a + 1 (b − e) (e − a) ( a − b ) (be – aa') (ca – bb') (ab — cc') Hm-» (c – abc m--3 m-39 where aa' = bb' = cc', and H is the sum of the homogeneous products of a, b, c, a', c', b' of m - 3 dimensions. 27. Shew that the product of any r consecutive terms of the series 1-c, 1− c2, 1 − c3,... is divisible by the first r of them. 28. Shew that, if c be numerically less than unity, (1 + cx) (1 + c2x) (1 + c3x)...to infinity 29. Shew that, if c be numerically less than unity, (1 + cx) (1 + c3x) (1 + c3x)..... to infinity |