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 (1+x)^n = 1 + nx + (n(n-1))/(2!)x^2 + with x=-y, n=(1)/(3). Step 1: Calculate terms. 1 (1)/(3)(-y) = -(1)/(3)y (1)/(3)((1)/(3)-1)2!(-y)^2 = (1)/(3)(-(2)/(3))2y^2 = (-2)/(9)2y^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 + 6x^-1 - x^-3 Step 2: Multiply by (x+1) and find x^-3 terms. (x+1)(8x^3 - 12x + 6x^-1 - x^-3) Terms contributing to x^-3: • x · (term with x^-4): None in the expansion. • 1 · (term with x^-3): 1 · (-x^-3) = -x^-3 Step 3: Identify the coefficient. The coefficient of x^-3 is -1. The final answer is -1.