Question

高数,泰勒公式,麦克劳林公式,急求高手.

Answer 1

就是换元,令 u = x²,

e^u = 1+u + u²/2!+ u³/3!+ .+ u^n /n!+ o(x^n)

再代入 u = x²,这个是利用间接法把函数展成Maclaurin公式.

更简单的,1/(1-x) = 1+ x + x² + x³ + ,+ x^n + o(x^n)

1/(1-x²) = 1 + x² + x(²)²+ .+ x^(2n) + o(x^2n)

若令f(x) = e^(x²),f(0) = 1,

f'(x) = 2x * e^(x²),f'(0) = 0,

f''(x) = (2+4x²) * e^(x²),f''(0) = 2

f'''(x) = (12x + 8x³) * e^(x²),f'''(0) = 0

f''''(0) = 12 .