√e近似值 用高数知识

通过求f(x)=√e^x的n阶Maclaurin公式
因为f'(x)=1/2*√e^x，f''=(1/2)^2*√e^x，f'''=(1/2)^3*√e^x,....f^(x)=(1/2)^(n+1)*√e^x
f(0)=1
f'(0)=1/2
f''(0)=(1/2)^2
...
f^(0)=(1/2)^(n+1)
f(x)=f(0)+f'(0)x+f''(0)x^2/2!+f'''(0)x^3/3!+....+f^(0)x^n/n!+f^(ξ)x^(n+1)/(n+1)!
其中ξ介于0与x之间
√e^x=1+x/2+(1/2)^2x^2/2!+(1/2)^3x^3/3!+....+(1/2)^nx^n/n!+(1/2)^(n+1)√e^x*x^(n+1)/(n+1)!
取√e^x≈1+x/2+(1/2)^2x^2/2!+(1/2)^3x^3/3!+....+(1/2)^nx^n/n!
产生的误差|Rn(x)|=(1/2)^(n+1)√e^x*x^(n+1)/(n+1)!
因为这个误差极小，所以可以忽略
当x=1时
√e≈1+1/2+(1/2)^2/2!+(1/2)^3/3!+...+(1/2)^n/n!
若取n=4
√e≈1+1/2+(1/2)^2/2!+(1/2)^3/3!+(1/2)^4/4!
√e≈1+0.5+0.125+0.0208+0.0026
√e≈1.6484
n取值越大，误差越小

因为e约等于2.718,与3接近,√3约等于1.732
√e=√(3-(3-e))=√3*(1-(3-e)/3)^(1/2)
=√3*(1-(1/2)*(3-e)/3+1/8*{(3-e)/3}^2+.
=1.732*(1-0.047+0.001)=1.732*0.954=1.6523