AMAT $220$ Project

The project is to estimate $$ using power series that we learned in class. Some problems are to be done manually, some by using a programming language $($Python preferred$)$ or computer software program of your choice $($e$.$g$.$ Excel, Matlab.

You can have up to two collaborators and should list them in the report.

Your report should be submitted on Canvas as a single PDF document, due no later than December $16,11$:$59$ pm$.$

Answer the following questions. When you do the work manually, please write legibly

Find a power series expansion centered at $x=0$ for $arctanx$ and find the interval of convergence of the above power series.

Express arctan $\frac{1}{\sqrt[\phantom{2}]{3}}$ as an alternating series ${\displaystyle \underset{k=0}{\overset{}{}}}(-1{)}^k{a}_k,$ then use the Alternating Series Estimation Theorem $($the exact formulation differs from book to book, but all the versions reach the same conclusion$)$ to find the smallest $N$ that estimate $S_N,$ the $N$ th partial sum of the series, to $$ within ${10}^{-6}.$ Recall that $\frac{}{6}=arctan(\frac{1}{\sqrt[\phantom{2}]{3}})$ and $=6*\frac{}{6}.$

Veriry your work in Part $2$ by explicityly calculating $S_n$ and $|S_n-|$ for $1nN+2,$ where $N$ is the number you found in Part $3.$ Save all the outputs in an appropritae format $($screenshot$,$ print as pdf etc.$)$ and include as part of your work.

Repeat Parts $2$ and $3$ with $\frac{}{4}=arctan1$ and $=4*\frac{}{4}.$ More precisely, express $arctan1$ as an alternating series ${\displaystyle \underset{k=0}{\overset{}{}}}(-1{)}^k{b}_k$ and find the smallest $M$ that estimate $S_M,$ the $M$ th partial sum of the series, to $$ within ${10}^{-2}.$ Then veriry your work by explicityly calculating $S_n$ and $|S_n-|$ for $1nM+2.$ Save all the outputs in appropritae format $($screenshot$,$ print as pdf etc.$)$ and include as part of your work.

Can you use $\frac{}{3}=arctan\sqrt[\phantom{2}]{3}$ and $=3*\frac{}{3}$ to estimate $$ within the accuracy of your choice? If yes, then estimate $$ within $0\mathrm{.}01$ accuracy. If no$,$ explain why.

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