The Taylor series about $x=0$ for inverse tangent function is given by

$ta{n}^{-1}(x)={\displaystyle \underset{n=0}{\overset{}{}}}\frac{(-1{)}^n{x}^{2n+1}}{2n+1}$

$($a$)$ Write a computer program to evaluate $ta{n}^{-1}(x)$ by truncating the series to $N+1$ terms. $($Make sure you include your code in the submission.$)$ Describe the choices you made in writing your program.

$($b$)$ Use your program to approximate $$ by evaluating $ta{n}^{-1}(1)$ for different values of $N.$ Display your results in a form of a table. Make sure your table also includes absolute and relative errors. For error calculation use that $$~~$3\mathrm{.}141592653589793.$

$($c$)$ Repeat the same work by using $ta{n}^{-1}(\frac{\sqrt[\phantom{2}]{3}}{3}).$

$($d$)$ Plot the errors and approximations from parts $($a$)$ and $($b$)$ versus the number of terms in the Taylor expansion on a semilog $($log$-$linear$)$ and log$-$log graph. Look at the graphs you made and make conclusions about the rate of convergence. $($In MATLAB you can use commands semilogy and loglog$.)$