Chapter 7: Q7P (page 378)

Use Parseval’s theorem and the results of the indicated problems to find the sum of the series in Problems 5 to 9. The series $\sum_{n = 1}^{\infty} \frac{1}{n^{2}}$ ,using problem 5.8.

Short Answer

Expert verified

By Parseval theorem $\frac{π^{2}}{6} = \sum_{n = 1}^{\infty} \frac{1}{n^{2}}$

Step by step solution

Given Information.

The given series is $\sum_{n = 1}^{\infty} \frac{1}{n^{2}}$ .The sum of the series is to be found out.

Definition of Parseval’s theorem

According to Parseval's Theorem, a signal's energy can be defined as the average energy of its frequency components

Sum of the series

It is know that the solution of the problem 5.8 is $f (x) = 1 - 2 \sum_{n = 1}^{\infty} \frac{\sin (n x)}{n} {(- 1)}^{n}$

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

$< {|f (x)|}^{2} > = \frac{1}{2 π} \int_{- π}^{π} {(1 + x)}^{2} d x = \frac{1}{2 π} {(1 + x)}^{3} {|^{π}}_{- π} = 1 + \frac{π^{2}}{3}$

Use the Parseval theorem

$1 + \frac{π^{2}}{3} = 1 + \frac{1}{2} \sum_{n = 1}^{\infty} {(\frac{2 {(- 1)}^{n}}{n})}^{2} \Rightarrow \frac{π^{2}}{6} = \sum_{n = 1}^{\infty} \frac{1}{n^{2}}$

Thus $\frac{π^{2}}{6} = \sum_{n = 1}^{\infty} \frac{1}{n^{2}}$

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Use Parseval’s theorem and the results of the indicated problems to find the sum of the series in Problems 5 to 9. The series $\sum_{n = 1}^{\infty} \frac{1}{n^{2}}$ ,using problem 5.8.

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 Problems 5 to 9. The series ∑n=1∞1n2,using problem 5.8.

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

Force

Physics of Motion

Quantum Physics

Further Mechanics and Thermal Physics

Circular Motion and Gravitation

Electrostatics

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 Problems 5 to 9. The series $\sum_{n = 1}^{\infty} \frac{1}{n^{2}}$ ,using problem 5.8.