This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions. 3. Find the first three terms in ascending powers of y in the expansion of (1-y)^(1)/(3). Use the binomial expansion (1+x)^n = 1 + nx + (n(n-1))/(2!)x^2 + Here x=-y and n=(1)/(3). Step 1: First term. 1 Step 2: Second term. n(-y) = (1)/(3)(-y) = -(1)/(3)y Step 3: Third term. (n(n-1))/(2!)(-y)^2 = (1)/(3)((1)/(3)-1)2 y^2 = (1)/(3)(-(2)/(3))2 y^2 = (-2)/(9)2 y^2 = -(1)/(9)y^2 The first three terms are 1 - (1)/(3)y - (1)/(9)y^2. 4. Find the coefficient of x^-3 in the expansion of ((x+1)(2x - (1)/(x))^3). Step 1: Expand (2x - (1)/(x))^3. (2x - (1)/(x))^3 = (2x)^3 + 3(2x)^2(-(1)/(x)) + 3(2x)(-(1)/(x))^2 + (-(1)/(x))^3 = 8x^3 - 12x + (6)/(x) - (1)/(x^3) = 8x^3 - 12x + 6x^-1 - x^-3 Step 2: Multiply by (x+1). (x+1)(8x^3 - 12x + 6x^-1 - x^-3) = x(8x^3 - 12x + 6x^-1 - x^-3) + 1(8x^3 - 12x + 6x^-1 - x^-3) = (8x^4 - 12x^2 + 6x^0 - x^-2) + (8x^3 - 12x + 6x^-1 - x^-3) Step 3: Identify the coefficient of x^-3. The only term with x^-3 is -x^-3. The coefficient of x^-3 is -1. The final answer is -1.