Suppose that f(x) and g(x) are given by the power series f(x) = 7+ 4x + 3x² + 3x³ +... and g(x) = 35+34x + 51x² +51x³+.... By dividing power series, find the first few terms of the series for the quotient h(x) = = co+c₁x + ₂x² + 3x³ +.... g(x) f(x) 11 C1 = S 11 C2= 3 ||

1. The function f(x) satisfies: f(0) = 0; f'(0) = -1; f" (0) = 4; f"" (0) = 54, and f(4) (0) = 8. Find the first four non-zero terms of the Maclaurin series for this function. OA. -x+4x2+54x³+8x4 OB. None of these OC.-x+ 2x² + 9x³ + x4 OD. -1+ 4x + 27x² + x³ OE.-x+ 2x²+18x³ + 2x4

Suppose that f(x) and g(x) are given by the power series f(x) = 2 + 6x + 4x² + 3x³ + ... and g(x) = 4 + 8x+3x²+2x³ +. By multiplying power series, find the first few terms of the series for the product h(x) = f(x) · g(x) = co + c₁x + c₂x² + 3x³ + .... Co = C1 = || S C3 || 1-0

4. Find the function given by the power series n x+2 (x + ²)", when -7 < x < 3. 5 f(x) = Σ n="