(a) Evaluate the integral 2 32 -dx. x² +4 Your answer should be in the form kл, where k is an integer. What is the value of k? Hint: d dx 1 -arctan(2) = x²+1 k = (b) Now, lets evaluate the same integral using power series. First, find the power series for the function f(x): 32 Then, integrate it from 0 to 2, and call it S. S should be an infinite series. x²+4 What are the first few terms of S? ao a1 a2 a3 a4 (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.)
(a) Evaluate the integral 2 32 -dx. x² +4 Your answer should be in the form kл, where k is an integer. What is the value of k? Hint: d dx 1 -arctan(2) = x²+1 k = (b) Now, lets evaluate the same integral using power series. First, find the power series for the function f(x): 32 Then, integrate it from 0 to 2, and call it S. S should be an infinite series. x²+4 What are the first few terms of S? ao a1 a2 a3 a4 (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.)
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.2: Arithmetic Sequences
Problem 68E
Related questions
Question
![(a) Evaluate the integral
2
32
-dx.
x² +4
Your answer should be in the form kл, where k is an integer. What is the value of k?
Hint:
d
dx
1
-arctan(2)
=
x²+1
k
=
(b) Now, lets evaluate the same integral using power series. First, find the power series for the function
f(x):
32
Then, integrate it from 0 to 2, and call it S. S should be an infinite series.
x²+4
What are the first few terms of S?
ao
a1
a2
a3
a4
(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.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F953e3ae3-1a36-4eca-a542-ddeb8c6ae72a%2F428bcf6e-9a7d-4a11-8e0a-529a6fe807ce%2Fcog0ce_processed.png&w=3840&q=75)
Transcribed Image Text:(a) Evaluate the integral
2
32
-dx.
x² +4
Your answer should be in the form kл, where k is an integer. What is the value of k?
Hint:
d
dx
1
-arctan(2)
=
x²+1
k
=
(b) Now, lets evaluate the same integral using power series. First, find the power series for the function
f(x):
32
Then, integrate it from 0 to 2, and call it S. S should be an infinite series.
x²+4
What are the first few terms of S?
ao
a1
a2
a3
a4
(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.)
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