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Σ (-1)"x2n (2n)! 1. For all x E R, cos x = n=0 (a) Find a power series that is equal to a cos(x²) for all a e R. (b) Differentiate the series in (la) to find a power series that is equal to cos(x²) – 2x2 sin(x²) for all r E R. (c) Use the result in (1b) to prove that S-16)"(4n + 1) (2n)! +oo cos(4) – 8 sin(4). %3D n=0

Σ (-1)"x2n (2n)! 1. For all x E R, cos x = n=0 (a) Find a power series that is equal to a cos(x²) for all a e R. (b) Differentiate the series in (la) to find a power series that is equal to cos(x²) – 2x2 sin(x²) for all r E R. (c) Use the result in (1b) to prove that S-16)"(4n + 1) (2n)! +oo cos(4) – 8 sin(4). %3D n=0

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.3: Geometric Sequences
Problem 50E
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+oo (-1)"2n 1. For all x € R, cosx = Σ (2n)! n=0 (a) Find a power series that is equal to a cos(x2) for all a 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 Σ (-16)"(4n + 1) (2n)! cos(4) – 8 sin(4). n=0
Transcribed Image Text:+oo (-1)"2n 1. For all x € R, cosx = Σ (2n)! n=0 (a) Find a power series that is equal to a cos(x2) for all a 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 Σ (-16)"(4n + 1) (2n)! cos(4) – 8 sin(4). n=0
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