Chapter 7: Q9P (page 378)

Use Parseval’s Theorem and the results of the indicated problems to find the sum of the series in Problems 5to 9
The series $\frac{1}{3^{2}} + \frac{1}{15^{2}} + \frac{1}{35^{2}} + . . .$ , using Problem 5.11

Short Answer

Expert verified

The value of the series $\frac{1}{3^{2}} + \frac{1}{15^{2}} + \frac{1}{35^{2}} + . . .$ is $\frac{π^{2}}{16} - \frac{1}{2}$ by the use of Parseval’s theorem

Step by step solution

Definition of Parseval’s Theorem

In mathematics, Parseval's theorem generally refers to the result that the Fourier transfigure is unitary; approximately, that the sum of the forecourt of a function is equal to the sum of the forecourt of it is transfigure.

Step 2: Given Parameters

Given the series $\frac{1}{3^{2}} + \frac{1}{15^{2}} + \frac{1}{35^{2}} + . . .$

Value of the series is to be found using Parseval Theorem and Problem 5.11.

Use the function from problem 5.11

From Problem 5.11, write the function as $f (x) = \frac{1}{π} + \frac{1}{2} \sin x - \frac{2}{π} \sum_{n - e v e n}^{\infty} \frac{\cos (n x)}{n^{2} - 1}$ where $n \neq 0$

Find the mean value of square of the function

$<f^{2} (x)> = \frac{1}{2 π} \int_{0}^{π} \sin^{2} x d x = \frac{1}{4 π} \int_{0}^{π} (1 - \cos (2 x)) d x = \frac{1}{4}$

Now, use the Parseval Theorem to get

$\frac{1}{4} = \frac{1}{π^{2}} + \frac{1}{2} {(\frac{1}{2})}^{2} - \frac{1}{2} \sum_{n - e v e n}^{\infty} {(\frac{2}{π (n^{2} - 1)})}^{2}$

Solve further to get

$\sum_{n - e v e n}^{\infty} \frac{1}{{(n^{2} - 1)}^{2}} = \frac{π^{2}}{16} - \frac{1}{2}$

Therefore, the sum of series $\frac{1}{3^{2}} + \frac{1}{15^{2}} + \frac{1}{35^{2}} + . . .$ is $\frac{π^{2}}{16} - \frac{1}{2}$ by the use of Parseval’s theorem.

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Use Parseval’s Theorem and the results of the indicated problems to find the sum of the series in Problems 5to 9
The series $\frac{1}{3^{2}} + \frac{1}{15^{2}} + \frac{1}{35^{2}} + . . .$ , using Problem 5.11

Short Answer

Step by step solution

Definition of Parseval’s Theorem

Step 2: Given Parameters

Use the function from problem 5.11

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Use Parseval’s Theorem and the results of the indicated problems to find the sum of the series in Problems 5to 9The series 132+1152+1352+..., using Problem 5.11

Short Answer

Step by step solution

Definition of Parseval’s Theorem

Step 2: Given Parameters

Use the function from problem 5.11

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

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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 5to 9
The series $\frac{1}{3^{2}} + \frac{1}{15^{2}} + \frac{1}{35^{2}} + . . .$ , using Problem 5.11