Question

f(x)=arcsin x 带Peano余项的3阶Maclaurin公式是

Answer 1

f'(x) = 1/√(1-x²)

f''(x) = [-1/(1-x²)] * [1/(2√(1-x²))] * (-2x) = x/(1-x²)^(3/2)

f'''(x) = [(1-x²)^(3/2) - x * (3/2 (1-x²)^(1/2) * (-2x)]/(1-x²)³

f(x)带Peano余项的3阶Maclaurin公式其实就是f(x)在0点的带Peano余项的3阶Taylo展开式,即

f(x) = f(0) + f'(0)x + f''(0)x²/2 + f'''(0)x³/6 + o(x³)

= 0 + 1x + 0x²/2 + x³/6 + o(x³)

= x + x³/6 + o(x³)