泰勒展开式利用python数值方法证明

from sympy import *
from sympy import log,sin,exp
import math #定义变量为x
x=Symbol("x") #函数为

def taylor (f = x**4,n = 10,x0 = 0,t = 2):

    #n = 100 #泰勒展开项数
    i = 0
    F = list()
    #n阶导
    while i <= n:
        f1 = diff(f,x,i)
        value = f.evalf(subs={x: x0})
        F.append(value)
        i = i+1
    print (F)
    #求阶乘
    M = list()
    for i in range (n+1):
        if i == 0:
            M.append(1)
        else:
            for j in range(i):

                if j == 0:
                    v = 1
                else:
                    v = v*(j+1)
            M.append(v)
    print (M)


    #求(x - x0)^i
    J = list()
    for i in range(n+1):
        v = (t - x0)**i
        J.append(v)
    print (J)
    V = list()
    for i in range(n+1):
        value = J[i]/M[i]*F[i]
        V.append(value)
    return sum(V)
def main():
    f = exp(x)

    n = int(input('请输入泰勒展开项数'))
    x0 = int(input('请输入x0'))
    t = int(input('请输入x'))
    value = f.evalf(subs={x: t})
    predict = taylor(f,n,x0,t)
    print (value)
    print (predict)

main()

 默认f(x) = exp(x)

请输入泰勒展开项数10
请输入x02
请输入x9
[7.38905609893065, 7.38905609893065, 7.38905609893065, 7.38905609893065, 7.38905609893065, 7.38905609893065, 7.38905609893065, 7.38905609893065, 7.38905609893065, 7.38905609893065, 7.38905609893065]
[1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800]
[1, 7, 49, 343, 2401, 16807, 117649, 823543, 5764801, 40353607, 282475249]
8103.08392757538
7304.76166428328

 

请输入泰勒展开项数10
请输入x05
请输入x9
[148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577]
[1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800]
[1, 4, 16, 64, 256, 1024, 4096, 16384, 65536, 262144, 1048576]
8103.08392757538
8080.07306436618

可见泰勒公式的近似值与x0无关

/usr/bin/python3.5 /home/rui/ComputerScience/PythonProjects/数值分析/Taylor.py
请输入泰勒展开项数100
请输入x05
请输入x9
[148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577, 148.413159102577]
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8103.08392757538
8103.08392757538

 当项数达到100时可见已经十分逼近正确值