Question

The Taylor series expansion of sin x about x=0 is

sin x = x-x³/3!+x⁵/5!-x⁷/7!+x⁹/9!-...
Write a computer program which estimates sin x at x = π/4 It should include every term with a magnitude larger than ϵ = 10⁻¹².In your program, define and properly use a function called 'Factorial' to calculate the Factorial of an integer. Output your estimate and its difference from the correct answer supplied directly by the built-in sine function.

Answer 1

The student's question involves writing a program to estimate sin x using its Taylor series expansion about x = 0, with a precision threshold for the terms. The Factorial function is essential for computing the series terms, and the estimation is compared to the math library's sine function for accuracy.

Step-by-step explanation:

A student needs to write a computer program to estimate sin x at x = π/4 using its Taylor series expansion, considering terms with a magnitude larger than ε = 10-12. To achieve this, the student is instructed to define a function called 'Factorial' to compute factorials, which are essential in calculating the terms of the Taylor series.

Here is a pseudo-code like example which outlines how the program can be structured:

def Factorial(n):
if n == 0 or n == 1:
return 1
else:
return n * Factorial(n - 1)

x = π/4
ε = 10**(-12)
sin_estimated = x
term = x
i = 1

while abs(term) > ε:
i += 2
term = (-1)**((i - 1)/2) * (x**i)/Factorial(i)
sin_estimated += term

sin_true = math.sin(x)
difference = abs(sin_true - sin_estimated)
print('Estimated sin(x):', sin_estimated)
print('Difference:', difference)

This pseudo-code initializes the estimate with the first term of the Taylor series, and iteratively computes subsequent terms until they are smaller than ε. The Factorial function is a simple recursive calculation of n!. The true sine value and the difference between the estimate and the true value are then calculated and printed.