(a) Evaluate the integral
$\displaystyle \int_{0}^{2} \frac{24}{x^2+4} dx$.
Your answer should be in the form $k\pi$, where $k$ is an integer. What is the value of $k$?
(Hint: $\frac{d \arctan(x)}{dx} = \frac{1}{x^2+1}$ )
$k =$
(b) Now, lets evaluate the same integral using power series. First, find the power series for the function $f(x) = \frac{24}{x^2+4}$. Then, integrate it from 0 to 2, and call it S. S should be an infinite series $\sum_{n=0}^\infty a_n$ .
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 $\pi$ in terms of an infinite series. Approximate the value of $\pi$ by the first 5 terms.
.
(d) What is an upper bound for your error of your estimate if you use the first 10 terms? (Use the alternating series estimation.)
.