(a) Evaluate the integral: S Hint: Your answer should be in the form ki, where k is an integer. What is the value of k? d 1 -arctan(r) dx ²+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)= x²+4 What are the first few terms of S? ao a1 a2 a3 || || 16 x² + 4 || || dr 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 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 11 terms? (Use the alternating series estimation.)
(a) Evaluate the integral: S Hint: Your answer should be in the form ki, where k is an integer. What is the value of k? d 1 -arctan(r) dx ²+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)= x²+4 What are the first few terms of S? ao a1 a2 a3 || || 16 x² + 4 || || dr 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 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 11 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: S
Hint: -arctan(r)
Your answer should be in the form kπ, where k is an integer. What is the value of k?
d
1
dx
x² +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) =
x² + 4
What are the first few terms of S?
ao
a1
a2
a3
a4
||
||
16
x² + 4
||
||
dr
||
(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 by the first 5 terms.
(d) What is the upper bound for your error of your estimate if you use the first 11 terms? (Use the
alternating series estimation.)
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