Problem $2.[35$ points$]$ The Maclaurin series of $sin(x)$ can be written as:

$sinx={\displaystyle \underset{n=0}{\overset{}{}}}\frac{(-1{)}^n}{(2n+1)!}{x}^{2n+1},=x-\frac{x^3}{3!}+\frac{x^5}{5!}-$cdots

Here the factorial of a positive integer $n!$ is defined as the product of all positive

integers less than or equal to $n.$

Write a C$++$ program that takes a double variable $x$ and returns the

approximation of $sin(x)$ using the first $k$ terms of the Maclaurin series. For

example, when $k=1,$ we have $sin(x)$~~$x.$ When $k=2,$ we have $sin(x)$~~$x-\frac{x^3}{3!}.$

The program should display the difference between your approximations and

$sin(x)$ in $k=1,2,3,$dots,$10$