Need help with only parts e & f. 5. If we know that F(x) =an(x – c)" for all |r – e| < R_for R > 0, then we can use differentiation,n-0integration and substitution to find new power series representations for F (x), | F(x) dx and F(g(x)).(a) Use the Geometric Series Test to find a power series representation centered at e = 0 for f(x) =11+x² '(b) Use term-by-term differentiation to find a power series representation centered at e = 0 of f'(x).Determine the radius of convergence and the interval of convergence.Determine the radius of convergence and the interval of convergence.1(c) Use part (b) to find a power series representation centered at e = 0 for(1 + x²)² °What is theradius of convergence and the interval of convergence for the power series?1(d) Use part (c) and substitution to find a power series representation centered at e= 0 of(1 + x*)² *Determine the radius of convergence and the interval of convergence.(e) Use term-by-term integration to find a power series representation centered at e = 0 fordr.What is the radius and interval of convergence for this series?r1/21(f) Find a series that represents I = (1+#^)2Theorem to find an approximation to the value of the definite integral I that is within 10-3 of theactual value. (Hint: What is the smallest N such that SN is guaranteed to be within 10–3 of thesum 1?)dr. Then use the Alternating Series Estimation

For part e), we first need to find the power series representation of 1(1+x4)2 around the point 0.…

1. (e) Use term-by-term integration to find a power series representation centered at e = 0 for / E dr. What is the radius and interval of convergence for this series? 1 (f) Find a series that represents I = 1+#*)2 dr. Then use the Alternating Series Estimation Theorem to find an approximation to the value of the definite integral I that is within 10-3 of the actual value. (Hint: What is the smallest N such that SN is guaranteed to be within 10-3 of the sum 1?)

1. (e) Use term-by-term integration to find a power series representation centered at e = 0 for / E dr. What is the radius and interval of convergence for this series? 1 (f) Find a series that represents I = 1+#*)2 dr. Then use the Alternating Series Estimation Theorem to find an approximation to the value of the definite integral I that is within 10-3 of the actual value. (Hint: What is the smallest N such that SN is guaranteed to be within 10-3 of the sum 1?)

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

Need help with only parts e & f.

5. If we know that F(x) =an(x – c)" for all |r – e| < R_for R > 0, then we can use differentiation,
n-0
integration and substitution to find new power series representations for F (x), | F(x) dx and F(g(x)).
(a) Use the Geometric Series Test to find a power series representation centered at e = 0 for f(x) =
1
1+x² '
(b) Use term-by-term differentiation to find a power series representation centered at e = 0 of f'(x).
Determine the radius of convergence and the interval of convergence.
Determine the radius of convergence and the interval of convergence.
1
(c) Use part (b) to find a power series representation centered at e = 0 for
(1 + x²)² °
What is the
radius of convergence and the interval of convergence for the power series?
1
(d) Use part (c) and substitution to find a power series representation centered at e= 0 of
(1 + x*)² *
Determine the radius of convergence and the interval of convergence.
(e) Use term-by-term integration to find a power series representation centered at e = 0 for
dr.
What is the radius and interval of convergence for this series?
r1/2
1
(f) Find a series that represents I = (1+#^)2
Theorem to find an approximation to the value of the definite integral I that is within 10-3 of the
actual value. (Hint: What is the smallest N such that SN is guaranteed to be within 10–3 of the
sum 1?)
dr. Then use the Alternating Series Estimation

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