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(-1)"x²n (2n)! 1. For all x € R, cos ax = n=0 (a) Find a power series that is equal to x cos(x²) for all x E R. (b) Differentiate the series in (la) to find a power series that is equal to cos(x2) – 2x? sin(x²) for all x E R. (c) Use the result in (1b) to prove that (-16)"(4n + 1) Σ cos(4) – 8 sin(4). (2n)! n=0 W!

(-1)"x²n (2n)! 1. For all x € R, cos ax = n=0 (a) Find a power series that is equal to x cos(x²) for all x E R. (b) Differentiate the series in (la) to find a power series that is equal to cos(x2) – 2x? sin(x²) for all x E R. (c) Use the result in (1b) to prove that (-16)"(4n + 1) Σ cos(4) – 8 sin(4). (2n)! n=0 W!

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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(-1)"x²n (2n)! 1. For all x € R, cos ax = n=0 (a) Find a power series that is equal to x cos(x²) for all x E R. (b) Differentiate the series in (la) to find a power series that is equal to cos(x²) – 2x² sin(x²) for all x E R. (c) Use the result in (1b) to prove that +oo Σ (-16)"(4n+ 1) (2n)! cos(4) – 8 sin(4). n=0 W!
Transcribed Image Text:(-1)"x²n (2n)! 1. For all x € R, cos ax = n=0 (a) Find a power series that is equal to x cos(x²) for all x E R. (b) Differentiate the series in (la) to find a power series that is equal to cos(x²) – 2x² sin(x²) for all x E R. (c) Use the result in (1b) to prove that +oo Σ (-16)"(4n+ 1) (2n)! cos(4) – 8 sin(4). n=0 W!
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