$3\mathrm{.}7$ Evaluate $e^{-3}$ using two approaches,

$e^{-x}=1-x+\frac{x^2}{2}-\frac{x^3}{3!}+$cdots

and

$e^{-x}=\frac{1}{e^x}=\frac{1}{1+x+\frac{x^2}{2}+\frac{x^3}{3!}+\text{c}\text{d}\text{o}\text{t}\text{s}}$

and compare with the true value of $6\mathrm{.}737947{10}^{-3}$ Use $20$ terms to evaluate each series and compute true and approximate relative errors as terms are added.

You will:

Create a Python file called A$1\_$ task $3.$ py that:

a$.$ Defines a Python function $($or two$)$ to approximate $e^{-x}$ using both approaches. Your function$($s$)$ should have at least two inputs: the value of $x($which will equal $5$ in this case$)$ and the number of terms to use, $n.$ In each of the approaches, $\text{'}1\text{'}$ counts as the first term.

b$.$ Uses the function$($s$)$ in part a$.$ to solve the textbook problem.

c$.$ Graphs the results $($approximation value, true percent relative error, and approximate percent relative error $(y-$axis$)$ as functions of the number of terms $(x$ axis$)$ used in the approximation$).$ Plots need to have sufficient resolution on the axes.

By hand, calculate true and approximate relative error $($expressed as percentages$)$ for both approaches using $1,2,$ and $3$ terms. Ensure your hand calculations match your Python results.

a$.$ Show your work for credit.

b$.$ Report true and approximate relative error $($expressed as percentages$)$ for both approaches in table form for the various number of terms $($i$.$e$.,1,2$ and $3)$ used.

Answer the following question: Of these two approximation methods, which one would you consider more desirable and why? Please explain. Refer to figures as necessary when answering.