Evaluate the integral...
int(0 to 2) of 40/(x^2 + 4)dx
a: Your answer should be in the form ` kpi `, where ` k ` is an integer. What is the value of ` k `?
b: Now, let's evaluate the same integral using power series. First, find the power series for the function ` f(x) = (40)/(x^2+4) `. Then, integrate it from 0 to 2, and call it S. S should be an infinite series.
What are the first few terms of S ?
c: The answers to part (a) and (b) are equal (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 the upper bound for your error of your estimate if you use the first 10 terms? (Use the alternating series estimation.)
The Attempt at a Solution
k is 5, this is correct, I know how to get k...
the first term of the series is 20, I know how to get that, but then the rest of my series goes wacky.
Here's my equation for the series
the sum of (-1)^n * ((2)^(2n + 1)/(10^(n - 1) * (2n + 1))
I can show how I got that in case anyone wants, but I think it's fairly obvious.
And at the end of that, using exact integral notation, there'd be the same equation but instead of that first 2 there'd be a 0, but that simplifies to 0 so it doesn't matter.
When I use n = 0, I get 20 like I'm supposed to, but when I use n = 1, i get -2.6667...answer is -6.6667
my next term is .64, should be getting 4...then -.182857, should get a -2.85714285714...then .0568888889, should get a 2.22222222222...
For c and d of course I'm going to be off, but I got 3.5694 and 3.64722E-6 respectively. The correct answers are 3.33968253968 and 0.190476190476.