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For all x € R, sin.x = Σ (−1)"x2n+1 (2n +1)!* n=0 ■Find a power series that is equal to x³ sin(x²) for all x € R. ■ Differentiate the series in item 1(a) to find a power series that is equal to 3x² sin(x²)+2x* cos(x²) for all x € R. 3 Use the result in item 1(b) to prove that Σ (−1)ª(4n+5) 16″+1(2n +1)! = ³² sin (1) += cos (1).

For all x € R, sin.x = Σ (−1)"x2n+1 (2n +1)!* n=0 ■Find a power series that is equal to x³ sin(x²) for all x € R. ■ Differentiate the series in item 1(a) to find a power series that is equal to 3x² sin(x²)+2x* cos(x²) for all x € R. 3 Use the result in item 1(b) to prove that Σ (−1)ª(4n+5) 16″+1(2n +1)! = ³² sin (1) += cos (1).

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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For all x € R, sin.x = Σ (-1)"x2n+1 no (2n+1)! ■Find a power series that is equal to x³ sin(x²) for all x € R. ■ Differentiate the series in item 1(a) to find a power series that is equal to 3x² sin(x²)+2x* cos(x²) for all x € R. Use the result in item 1(b) to prove that (-1)" (4n+5) 3 16"+1(2n +1)! n=0 = = sin()+cos (1).
Transcribed Image Text:For all x € R, sin.x = Σ (-1)"x2n+1 no (2n+1)! ■Find a power series that is equal to x³ sin(x²) for all x € R. ■ Differentiate the series in item 1(a) to find a power series that is equal to 3x² sin(x²)+2x* cos(x²) for all x € R. Use the result in item 1(b) to prove that (-1)" (4n+5) 3 16"+1(2n +1)! n=0 = = sin()+cos (1).
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