(a) Evaluate the integral: k = Your answer should be in the form kn, where k is an integer. What is the value of k? d 1 Hint: -arctan(2) dx x² +1 a1 = 2 32 x² 1²³ 0 What are the first few terms of S? ao = a2 = (b) Now, let's evaluate the same integral using a power series. First, find the power series for the function Then, integrate it from to 2, and call the result S. S should be an infinite series. 32 f(x) x² + 4 a3 = a4 = +4 dx (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 in terms of an infinite series. Approximate the value of π by the first 5 terms. (d) What is the upper bound for your error of your estimate if you use the first 7 terms? (Use the alternating series estimation.)
(a) Evaluate the integral: k = Your answer should be in the form kn, where k is an integer. What is the value of k? d 1 Hint: -arctan(2) dx x² +1 a1 = 2 32 x² 1²³ 0 What are the first few terms of S? ao = a2 = (b) Now, let's evaluate the same integral using a power series. First, find the power series for the function Then, integrate it from to 2, and call the result S. S should be an infinite series. 32 f(x) x² + 4 a3 = a4 = +4 dx (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 in terms of an infinite series. Approximate the value of π by the first 5 terms. (d) What is the upper bound for your error of your estimate if you use the first 7 terms? (Use the alternating series estimation.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Transcribed Image Text:(a) Evaluate the integral:
k
Your answer should be in the form kä, where k is an integer. What is the value of k?
1
Hint: -arctan(2)
d
dx
x² + 1
=
a1
(b) Now, let's evaluate the same integral using a power series. First, find the power series for the function
32
f(x)
Then, integrate it from 0 to 2, and call the result S. S should be an infinite series.
x² + 4
a2
What are the first few terms of S?
ao =
=
||
az =
2 32
x² + 4
=
=
a4 =
dx
(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 7 in terms of an infinite series. Approximate
the value of π by the first 5 terms.
(d) What is the upper bound for your error of your estimate if you use the first 7 terms? (Use the alternating
series estimation.)
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