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

tan$1($x$)=$

$\backslash$infty $$

n$=0$

$(1)$nx$2$n$+1$

$2$n $+1$

$($a$)$ Write a computer program to evaluate tan$1($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 $\backslash$pi by evaluating tan$1(1)$ 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 $\backslash$pi $3\mathrm{.}141592653589793.$

$($c$)$ Repeat the same work by using tan$1$

$(3$

$3$

$)$

$.$

$($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$.)$