# Plot Taylor polynomial approximations to sin x. The sine function can be approximated by a…

Plot Taylor polynomial approximations to sin x.

The sine function can be approximated by a polynomial

according to the following formula:

The expression (2j + 1)! is the factorial (see Exercise

3.14). The error in the approximation S(x; n) decreases as n increases and in

the limit we have that limn→∞ S(x; n) = sin x. The purpose of this

exercise is to visualize the quality of various approximations S(x; n) as n

increases.

The first part of the exercise is to write a Python function

S(x,

n) that computes S(x; n). Use a straightforward approach

where you compute each term as it stands in the formula, i.e., (−1)j_{x}^{2j+1} divided by the factorial (2j + 1)!. (We remark that Exercise

A.16 outlines a much more efficient computation of the terms in the series.)

The next part of the exercise is to plot sin x on [0,

4π] together with the approximations S(x; 1), S(x; 2), S(x; 3), S(x; 6),

and S(x; 12). Name of program file: plot_Taylor_sin.py.