1. Begin with the Taylor series cos x = 1 - 2! 4! + + and...
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1. Begin with the Taylor series cos x = 1 - 2! 4! + + and mimic Euler's work 6! (i.e. equate this infinite sum with an infinite product) to derive the sum of the recip- rocals of the squares of the odd integers: 1 1 1 1 1+=+ + + + ...= 9 25 49 81 77² 8 2. (a) Using the great theorem from this chapter and the result from question 1, evaluate 1 1 the alternating series 1 1 1 + + 4 9 16 25 (b) Use the product expension of cos x from question 1 to show √2 2 2 6 6 10 10 14 14... 1.3.5.7.9 11 13 15.. 1. Begin with the Taylor series cos x = 1 - 2! 4! + + and mimic Euler's work 6! (i.e. equate this infinite sum with an infinite product) to derive the sum of the recip- rocals of the squares of the odd integers: 1 1 1 1 1+=+ + + + ...= 9 25 49 81 77² 8 2. (a) Using the great theorem from this chapter and the result from question 1, evaluate 1 1 the alternating series 1 1 1 + + 4 9 16 25 (b) Use the product expension of cos x from question 1 to show √2 2 2 6 6 10 10 14 14... 1.3.5.7.9 11 13 15..
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Cos x can be expressed as the infinite product Cosx TT 1 4x n1 1472 1972 1 1 ... View the full answer
Related Book For
Finite Mathematics and Its Applications
ISBN: 978-0134768632
12th edition
Authors: Larry J. Goldstein, David I. Schneider, Martha J. Siegel, Steven Hair
Posted Date:
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