Chapter 7: Q6P (page 377)

Use Parseval’s theorem and the results of the indicated problems to find the sum of the series in Probllems 5 to 9. The series $\sum_{n = 1}^{\infty} \frac{1}{n^{4}}$ ,using problem 9.9.

Short Answer

Expert verified

By Parseval theorem $\frac{π^{4}}{90} = \sum_{n = 1}^{\infty} \frac{1}{n^{4}}$

Step by step solution

Given Information.

The given series is $\sum_{n = 1}^{\infty} \frac{1}{n^{4}}$ .The sum of the series is to be found out.

Definition of Parseval’s theorem

Parseval’s theorem is a theorem stating that the energy of a signal can be expressed as its frequency components’ average energy.

Sum of the series

It is know that the solution of the problem 9.9 is $f (x) = \frac{1}{12} + \frac{1}{π^{2}} \sum_{n = 1}^{\infty} \frac{\cos (2 π n x)}{n^{2}} {(- 1)}^{n}$

It is known that the Mean value of the square of the function is

$< {|f (x)|}^{2} > = \int_{- \frac{1}{2}}^{\frac{1}{2}} x^{4} d x = \frac{1}{5} x^{5} {|_{\frac{- 1}{2}}}^{\frac{1}{2}} = \frac{1}{80}$

Use the Parseval theorem

$\frac{1}{80} = \frac{1}{144} + \frac{1}{2} \sum_{n = 1}^{\infty} {(\frac{{(- 1)}^{n}}{π^{2} n^{2}})}^{2} \Rightarrow \frac{π^{4}}{90} = \sum_{n = 1}^{\infty} \frac{1}{n^{4}}$

Thus $\frac{π^{4}}{90} = \sum_{n = 1}^{\infty} \frac{1}{n^{4}}$

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Use Parseval’s theorem and the results of the indicated problems to find the sum of the series in Probllems 5 to 9. The series $\sum_{n = 1}^{\infty} \frac{1}{n^{4}}$ ,using problem 9.9.

Short Answer

Step by step solution

Given Information.

Definition of Parseval’s theorem

Sum of the series

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Use Parseval’s theorem and the results of the indicated problems to find the sum of the series in Probllems 5 to 9. The series ∑n=1∞1n4 ,using problem 9.9.

Short Answer

Step by step solution

Given Information.

Definition of Parseval’s theorem

Sum of the series

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Thermodynamics

Force

Wave Optics

Magnetism

Magnetism and Electromagnetic Induction

Measurements

Study anywhere. Anytime. Across all devices.

Use Parseval’s theorem and the results of the indicated problems to find the sum of the series in Probllems 5 to 9. The series $\sum_{n = 1}^{\infty} \frac{1}{n^{4}}$ ,using problem 9.9.