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can you please write each step clearly and on paper

can you please write each step clearly and on paper

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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can you please write each step clearly and on paper 

The function f(x) n=0 (-1)" x²n+1 (−1)" 22 = X = 2x has derivatives of all orders, and the Taylor series for f about x = 0 is 2 x² + 2 x³ x5 X + 2 7 x² X 4 8 a. Using the Ratio Test, determine the interval of convergence of the Taylor series about x = 0 for f. 2x 6. b. Let g(x) be the function such that g(0) = 0 and whose derivative is f(x) = 2 x + 2 Write the first four nonzero terms of the Taylor series for g(x) about x = 0. Use the first two nonzero terms of this series to estimate g 8 (²-). c. The series found in part (b) is an alternating series whose terms decrease in absolute value to 0. Show that the 1 approximation found in part (b) differs from g 1500 (=) by less than
Transcribed Image Text:The function f(x) n=0 (-1)" x²n+1 (−1)" 22 = X = 2x has derivatives of all orders, and the Taylor series for f about x = 0 is 2 x² + 2 x³ x5 X + 2 7 x² X 4 8 a. Using the Ratio Test, determine the interval of convergence of the Taylor series about x = 0 for f. 2x 6. b. Let g(x) be the function such that g(0) = 0 and whose derivative is f(x) = 2 x + 2 Write the first four nonzero terms of the Taylor series for g(x) about x = 0. Use the first two nonzero terms of this series to estimate g 8 (²-). c. The series found in part (b) is an alternating series whose terms decrease in absolute value to 0. Show that the 1 approximation found in part (b) differs from g 1500 (=) by less than
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