PROBLEM $2.$ points $40-$ Use Python

The power series for the function $arctan(x)$ is

$arctan(x)={\displaystyle \underset{i=1}{\overset{}{}}}(-1{)}^{i+1}\frac{x^{2i-1}}{2i-1}$

$($points $5)$ Show that $=4arctan(1).$ Using Python compute the absolute and relative errors between the exact value of $$ and its approximation $4arctan(1).$

$($points $30)$ Using the power series of $arctan(x),$ eq$.(1),$ find value $N$ so as the sum

$s_N={\displaystyle \underset{i=1}{\overset{N}{}}}(-1{)}^{i+1}\frac{x^{2i-1}}{2i-1}$

approximates the value of $\frac{}{4}$ with absolute error

Use a while $...$ break loop structure. After each new term is added, compute the absolute true error, the true percent relative error and the approximate percent relative error, eqs. $(3),(4)$ and $(5).$ Generate one plot for the value of $s_N$ versus number of terms added. Generate another plot for the absolute and approximate relative percent errors, eqs. $(3),(4)$ and $(5)$ respectively, versus number of terms added. Use a semilogarithmic scale for the $y-$axis of the errors plot.

$3.($points $5)$ Place the algorithm you wrote above in a Python function to compute an approximation of $$ using appropriate power series of $arctan(x).$