Homework $9$: Problem $11$

$(1$ point$)$

$($a$)$ Evaluate the integral

${\displaystyle \underset{0}{\overset{2}{}}}\frac{40}{x^2+4}dx$

Your answer should be in the form $k,$ where $k$ is an integer. What is the value of $k?$

$($Hint: $\frac{darctan(x)}{dx}=\frac{1}{x^2+1})$

$k=$

$($b$)$ Now, lets evaluate the same integral using power series. First, find the power series for the function $f(x)=\frac{40}{x^2+4}.$ Then, integrate it from $0$ to $2,$ and call it S$.$ S should be an infinite series ${\displaystyle \underset{n=0}{\overset{}{}}}{a}_n.$

What are the first few terms of $S?$

$a_0=$

$a_1=$

$a_2=$

$a_3=$

$a_4=$

$($c$)$ The answer in part $($a$)$ equals the sum of the infinite series in part $($b$)($why$?).$ Hence, if you divide your infinite series from $($b$)$ by $k($the answer to $($a$)),$ you have found an estimate for the value of $$ in terms of an infinite series. Approximate the value of $$ by the first $5$ terms.

$($d$)$ What is an upper bound for your error of your estimate if you use the first $11$ terms? $($Use the alternating series estimation.$)$

Note: You can earn partial credit on this oroblem.