将多项式1+2x+3x^2+4x^3+5x^4按(x+1)幂展开.

设f(x)=1+2x+3x^2+4x^3+5x^4,
taylor定理:
f(x)=f(x0)+f'(x0)(x-x0)+f''(x0)(x-x0)²/2!+……,
此处x0=-1,
f(x)=f(-1)+f'(-1)(x+1)+f''(-1)(x+1)²/2!+……
=(1+2x+3x^2+4x^3+5x^4)|(x=-1) + (2+6x+12x²+20x³)|(x=-1) × (x+1) + (6+24x+60x²)|(x=-1)
× (x+1)²/2 + (24+120x)|(x=-1) × (x+1)³/6 + 120×(x+1)⁴/24+0
=3-12(x+1)+21(x+1)²-16(x+1)³+5(x+1)⁴