Question

How to do part (c) and part(d）?

(a) Evaluate the integral 2 48 dx Your answer should be in the form kr, where k is an integer. What is the value of k? darctan(x dx r2+1 (b) Now, lets evaluate the same integral using power series. First, find the power series for the function f(x)-Then, integrate it from o to 2, and call it S. S should be an infinite series. What are the first few terms of S? :2+4 a224/5 a3-1-24/7 (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 π in terms of an infinite series. Approximate the value of by the first 5 terms (d) What is the upper bound for your error of your estimate if you use the first 6 terms? (Use the alternating series estimation.)

How to do part (c) and part(d）?

Answer 1

Solution :- C)  ∫(0 to 2) [48 / (x^2+4)] dx = 6π.

Therefore, 6π = Σ(n=0 to ∞) 24(-1)^n /(2n+1)
=> π = Σ(n=0 to ∞) 4(-1)^n /(2n+1).

Using the first five terms,
π ≈ 4(1 - 1/3 + 1/5 - 1/7 + 1/9) ≈ 3.33968253968.

d) Since the series for π is alternating, so using the alternating series estimation theorem , the error is no bigger than the absolute value of the next ( 6th) term, which is 4/13 ≈ 0.307692307

or error is 4/15 = 0.266666666667.