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1 Q.6 - Let S be the sum of the Basel series: S=1+ 25 49 81 1 1 1 1 + + + 25 9 16

1 Q.6 - Let S be the sum of the Basel series: S=1+ 25 49 81 1 1 1 1 + + + 25 9 16

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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Need answer of question 6 (as soon as possible)

Q..5 Begin with the Maclaurin (Taylor) series: cos: S2 = 1 + 2! 4! 6! (a) Evaluate cos 0. (b) List the zeros of cos 2. (c) Expand cos.r as an infinite product in the form P(x)=(1-) (1-) (1-7).... where a, b, c, ... are the zeros of cos r. (d) Equate the coefficients of the quadratic term in the Taylor expansion of cos to the corresponding coefficient in the infinite product representa- tion of cos r obtained in part (c). (e) Use the result in part (d) above to derive the sum of the reciprocals of the squares of the odd integers: 1 1 1 1 1+ + + + 9 25 49 81 Q.6 - Let S be the sum of the Basel series: 1 1 1 S = 1 + + + + (a) Show that the sum of the reciprocals of the squares of the even integers is given by: 1 1 1 1 1+ + + + + 4 16 36 64 (b) Combine the results above with the result from Question 5 to show that: 1 1 1 S = 1+ + 25 11
Transcribed Image Text:Q..5 Begin with the Maclaurin (Taylor) series: cos: S2 = 1 + 2! 4! 6! (a) Evaluate cos 0. (b) List the zeros of cos 2. (c) Expand cos.r as an infinite product in the form P(x)=(1-) (1-) (1-7).... where a, b, c, ... are the zeros of cos r. (d) Equate the coefficients of the quadratic term in the Taylor expansion of cos to the corresponding coefficient in the infinite product representa- tion of cos r obtained in part (c). (e) Use the result in part (d) above to derive the sum of the reciprocals of the squares of the odd integers: 1 1 1 1 1+ + + + 9 25 49 81 Q.6 - Let S be the sum of the Basel series: 1 1 1 S = 1 + + + + (a) Show that the sum of the reciprocals of the squares of the even integers is given by: 1 1 1 1 1+ + + + + 4 16 36 64 (b) Combine the results above with the result from Question 5 to show that: 1 1 1 S = 1+ + 25 11
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