(a) Evaluate the integral: Your answer should be in the form ka, where k is an integer. What is the value of k? d 1 Hint: -arctan(r) da 2² +1 k = (b) Now, let's evaluate the same integral using a power series. First, find the power series for the function Then, integrate it from 0 to 2, and call the result S. S should be an infinite series. 16 f(x) = 2² +4 What are the first few terms of S? ao a1 a2 || a4 || 16 x²+4 || a3 = || 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 by the first 5 terms.
(a) Evaluate the integral: Your answer should be in the form ka, where k is an integer. What is the value of k? d 1 Hint: -arctan(r) da 2² +1 k = (b) Now, let's evaluate the same integral using a power series. First, find the power series for the function Then, integrate it from 0 to 2, and call the result S. S should be an infinite series. 16 f(x) = 2² +4 What are the first few terms of S? ao a1 a2 || a4 || 16 x²+4 || a3 = || 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 by the first 5 terms.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Transcribed Image Text:(a) Evaluate the integral:
Hint:
=
Your answer should be in the form kn, where k is an integer. What is the value of k?
d
dx
—arctan(r)
a₁ =
a2 =
2 16
x² + 4
· 6²³
a3 =
(b) Now, let's evaluate the same integral using a power series. First, find the power series for the function
Then, integrate it from 0 to 2, and call the result S. S should be an infinite series.
16
f(x) =
x² + 4
What are the first few terms of S?
ao=
a4 =
dr
1
I²+1
(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 by the first 5 terms.
(d) What is the upper bound for your error of your estimate if you use the first 12 terms? (Use the
alternating series estimation.)
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