(a) Use the Geometric Series Test to find a power series representation centered at e = 0 for f(r) 1 Determine the radius of convergence and the interval of convergence. 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. (c) Use part (b) to find a power series representation centered at e = 0 for radius of convergence and the interval of convergence for the power series? 1 (1+ x²)2* What is the (d) Use part (c) and substitution to find a power series representation centered at e = 1 c = 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?
(a) Use the Geometric Series Test to find a power series representation centered at e = 0 for f(r) 1 Determine the radius of convergence and the interval of convergence. 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. (c) Use part (b) to find a power series representation centered at e = 0 for radius of convergence and the interval of convergence for the power series? 1 (1+ x²)2* What is the (d) Use part (c) and substitution to find a power series representation centered at e = 1 c = 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?
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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Need help with parts d -f, and with part d its okay if I just get the power series for it without using part c to find it.
Transcribed Image Text: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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