Taylor and Maclaurin series

Question

Use the definition of a Taylor series to find the first four nonzero terms of the series for f(x) centered at the given value of a. (Enter your answers as a comma-separated list.)

f(x) = 5 cos²(x),      a = 0

Answer 1

f(x) = 5/2 + 5/2cos2x; f(0) = 5; f'(x) = - 5sin2x; f'(0) = 0; f''(x) = -10cos2x; f''(x) = -10; f'''(x) = 20sin2x; f'''(0) = 0

f^iv = 40cos2x; f^iv(0) = 40; f^v = - 80sin2x; f^v(0) = 0; f^vi(x) = -160cos2x; f^vi(0) = -160

f(x) = 5 - 10x²/2! + 40x⁴/4! - 160x⁶/6! + ... = 5 - 5x² + 5/3x⁴ - 2/9x⁶ + ...