先来输出一下自然对数e:
>>>from math import e >>>print (e) 2.718281828459045理解e是复利增长的极限,有1块钱,复利100%,不管分成多少次,本利和都不会超过e≈2.718
泰勒展开式: f ( x ) = f ( x 0 ) + f ′ ( x 0 ) ( x − x 0 ) + 1 2 f ′ ′ ( x 0 ) ( x − x 0 ) 2 + . . . + f ( n ) ( x − x 0 ) n n ! + R n ( x ) f(x)=f(x_0)+f'(x_0)(x-x_0)+\frac{1}{2}f''(x_0)(x-x_0)^2+...+\frac{f^{(n)}(x-x_0)^n}{n!}+R_n(x) f(x)=f(x0)+f′(x0)(x−x0)+21f′′(x0)(x−x0)2+...+n!f(n)(x−x0)n+Rn(x) 其中n阶泰勒余项 R n ( x ) = f ( n + 1 ) ( ξ ) ( n + 1 ) ! ( x − x 0 ) n + 1 R_n(x)=\frac{f^{(n+1)}(\xi)}{(n+1)!}(x-x_0)^{n+1} Rn(x)=(n+1)!f(n+1)(ξ)(x−x0)n+1
令 f ( x ) = e x , x = 1 , x 0 = 0 f(x)=e^x,x=1,x_0=0 f(x)=ex,x=1,x0=0可得: e = 1 + 1 1 ! + 1 2 ! + . . . + 1 n ! + R ( n ) e=1+\frac{1}{1!}+\frac{1}{2!}+...+\frac{1}{n!}+R(n) e=1+1!1+2!1+...+n!1+R(n)
def factorial(n): result = 1 for i in range(1,n+1): result *= i return 1/result ee=1 for i in range(1,10): ee += factorial(i) print(ee)计算到第10项,可得e=2.7182815255731922,已经非常吻合。
可进一步了解: https://en.wikipedia.org/wiki/E_(mathematical_constant)