Problem $1)$ The Taylor series expansion of $ta{n}^{-1}x$ for $|x|<mn\; id="EditorElement\_318482">1</mn>$ is:

$ta{n}^{-1}x=x-\frac{x^3}{3}+\frac{x^5}{5}-$cdots$={\displaystyle \underset{n=0}{\overset{}{}}}(-1{)}^n\frac{x^{2n+1}}{2n+1}$

Develop an algorithm that performs the following $($you can present it in the form of a pseudo$-$code or a list of steps$)$:

The algorithm determines $ta{n}^{-1}x$ using the above equation, where the argument $x$ is the input, and the output argument $y$ is the approximate value for $ta{n}^{-1}x$ obtained using Taylor's series expansion. If $a_n$ is the $n$th term of the series, then sum $S_n$ of the first $n$ terms is $S_n=S_{n-1}+a_n.$ In each pass, calculate the estimated error $E$ given by $E=|a_n|.$ Stop adding terms when and report y$.$

Note: You are not asked to submit any code for this part. Submitting the algorithm is sufficient.