Find the Taylor series approximations

Use zero- through fourth-order Taylor series expansions to predict f(2.5) for f(x) = lnx using a base point at x = 1. Compute the true percent relative error E, for each approximation.

import math

# task info: x = 2.5 and x0 = 1

def func(x):

return math.log(x)

def nth_derivative(func, x, n, h):

"""

Approximate the nth derivative using central differences.

"""

if n == 0:

return func(x)

elif n == 1:

return (func(x + h) - func(x - h)) / (2 * h)

else:

# Recursive differentiation

return (nth_derivative(func, x + h, n - 1, h) -

nth_derivative(func, x - h, n - 1, h)) / (2 * h)

# Print derivatives from 0-th to 4-th order

for i in range(5):

print(i, "-th order derivative \t",

nth_derivative(func, x=1.0, n=i, h=0.001))

def taylor_series(func, x0, order, x_value):

# order - Taylor polynomial degree

if not callable(func):

raise TypeError("func must be a callable function.")

if not isinstance(order, int) or order < 0:

raise ValueError("order must be a non-negative integer.")

approx = 0.0

for n in range(order + 1):

# Compute nth derivative at x0 using numerical differentiation

derivative = nth_derivative(func, x0, n, h=0.001)

approx += (derivative / math.factorial(n)) * (x_value - x0) ** n

return approx

print()

print("Solution:")

x0 = 1

x_value = 2.5