k 2 x² + 4 Your answer should be in the form kä, where k is an integer. What is the value of k? (Hint: d arctan() dx = 10 ao a1 a2 аз || 1 || 40 (b) Now, lets evaluate the same integral using power series. First, find the power series for the function f(x) . Then, integrate it from 0 to 2, and call it S. S should be 40 x² +4 || = a) Evaluate the integral an infinite series. What are the first few terms of S ? dx. = = 1 x² +1 ) = 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 π by the first 5 terms. 0. (d) What is the upper bound for your error of your estimate if you use the first 12 terms? (Use the alternating series estimation.)
k 2 x² + 4 Your answer should be in the form kä, where k is an integer. What is the value of k? (Hint: d arctan() dx = 10 ao a1 a2 аз || 1 || 40 (b) Now, lets evaluate the same integral using power series. First, find the power series for the function f(x) . Then, integrate it from 0 to 2, and call it S. S should be 40 x² +4 || = a) Evaluate the integral an infinite series. What are the first few terms of S ? dx. = = 1 x² +1 ) = 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 π by the first 5 terms. 0. (d) What is the upper bound for your error of your estimate if you use the first 12 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
Related questions
Question
Transcribed Image Text:k
2
x² + 4
0
Your answer should be in the form kл, where k is an integer. What is the value of k?
(Hint:
)
=
a 1
a₂ =
az
a 4
a) Evaluate the integral
/\
=
40
10
(b) Now, lets evaluate the same integral using power series. First, find the power series
for the function f(x) =
=
Then, integrate it from 0 to 2, and call it S. S should be
=
dx.
=
d arctan(r)
dx
an infinite series.
What are the first few terms of S ?
ao =
=
1
x²+1
40
x²+4
(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 12 terms?
(Use the alternating series estimation.)
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