(a) Evaluate the integral: k= Your answer should be in the form kπ, where k is an integer. What is the value of k? d Hint: arctan(r) dx = a1 = A2 = S = (b) Now, let's evaluate the same integral using a power series. First, find the power series for th Then, integrate it from 0 to 2, and call the result S. S should be an infinite se 32 f(x) = x² +4 a3 = 32 x² + 4 What are the first few terms of S? ao = a4= 1 x² +1 dx (c) The answers to part (a) and (b) are equal (why?). Hence, if you divide your infinite series from (the answer to (a)), you have found an estimate for the value of π in terms of an infinite series. Ap 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 9 terms? (Use the alt- series estimation.)
(a) Evaluate the integral: k= Your answer should be in the form kπ, where k is an integer. What is the value of k? d Hint: arctan(r) dx = a1 = A2 = S = (b) Now, let's evaluate the same integral using a power series. First, find the power series for th Then, integrate it from 0 to 2, and call the result S. S should be an infinite se 32 f(x) = x² +4 a3 = 32 x² + 4 What are the first few terms of S? ao = a4= 1 x² +1 dx (c) The answers to part (a) and (b) are equal (why?). Hence, if you divide your infinite series from (the answer to (a)), you have found an estimate for the value of π in terms of an infinite series. Ap 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 9 terms? (Use the alt- 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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Transcribed Image Text:(a) Evaluate the integral:
k=
Your answer should be in the form kπ, where k is an integer. What is the value of k?
d
Hint: arctan(2)
dx
a1 =
a2 =
a3 =
1²
=
(b) Now, let's evaluate the same integral using a power series. First, find the power series for the fur
Then, integrate it from 0 to 2, and call the result S. S should be an infinite series.
32
f(x) =
x² +4
a4=
32
x² + 4
What are the first few terms of S?
ao =
1
x² +1
dx
(c) The answers to part (a) and (b) are equal (why?). Hence, if you divide your infinite series from (b) by
(the answer to (a)), you have found an estimate for the value of 7 in terms of an infinite series. Approxi
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 9 terms? (Use the alternatī
series estimation.)
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