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

Plot Taylor polynomial approximations to sin x.

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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)jx2j+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.