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Find the power series representation for g centered at 0 by differentiating or integrating the power series for f (perhaps more than once). Give the interval of convergence for the resulting series. g(x) = X (1+4x²)² using f(x) = 1 1+4x²

Find the power series representation for g centered at 0 by differentiating or integrating the power series for f (perhaps more than once). Give the interval of convergence for the resulting series. g(x) = X (1+4x²)² using f(x) = 1 1+4x²

Advanced Engineering Mathematics
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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Find the power series representation for g centered at 0 by differentiating or integrating the power series for f (perhaps more than once). Give the interval of convergence for the resulting series. g(x) = X k = 0 2 (1+4x²)² using f(x) = g(x)= (-1) 4k (k+1)x²k+1 1 1+ 4x² The power series g(x) converges on the interval (- 1 1 - 2'2 (Simplify your answer. Type your answer in interval notation. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.)
Transcribed Image Text:Find the power series representation for g centered at 0 by differentiating or integrating the power series for f (perhaps more than once). Give the interval of convergence for the resulting series. g(x) = X k = 0 2 (1+4x²)² using f(x) = g(x)= (-1) 4k (k+1)x²k+1 1 1+ 4x² The power series g(x) converges on the interval (- 1 1 - 2'2 (Simplify your answer. Type your answer in interval notation. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.)
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