5. If we know that F(x) = Lan(x – c)" for all |r – e| < R for R> 0, then we can use differentiation, integration and substitution to find new power series representations for F'(x), | F(x) dr and F(9(x)). (a) Use the Geometric Series Test to find a power series representation centered at e = 0 for f(x) = 1 1+z²* (b) Use term-by-term differentiation to find a power series representation centered at c = 0 of f'(1). 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 What is the (1+x²)2 ° radius of convergence and the interval of convergence for the power series?

5. If we know that F(x) = Lan(x – c)" for all |r – e| < R for R> 0, then we can use differentiation, integration and substitution to find new power series representations for F'(x), | F(x) dr and F(9(x)). (a) Use the Geometric Series Test to find a power series representation centered at e = 0 for f(x) = 1 1+z²* (b) Use term-by-term differentiation to find a power series representation centered at c = 0 of f'(1). 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 What is the (1+x²)2 ° radius of convergence and the interval of convergence for the power series?

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

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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