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Chapter 7: Q6P (page 377)

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

Short Answer

Expert verified

By Parseval theorem $\frac{蟺^{4}}{90} = {鈭�}_{n = 1}^{鈭�} \frac{1}{n^{4}}$

Step by step solution

01

Given Information.

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

02

Definition of Parseval’s theorem

Parseval鈥檚 theorem is a theorem stating that the energy of a signal can be expressed as its frequency components鈥� average energy.

03

Sum of the series

It is know that the solution of the problem 9.9 is $f (x) = \frac{1}{12} + \frac{1}{蟺^{2}} {鈭�}_{n = 1}^{鈭�} \frac{\cos (2 蟺 n x)}{n^{2}} {(- 1)}^{n}$

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

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

Use the Parseval theorem

$\frac{1}{80} = \frac{1}{144} + \frac{1}{2} {鈭�}_{n = 1}^{鈭�} {(\frac{{(- 1)}^{n}}{蟺^{2} n^{2}})}^{2} 鈬� \frac{蟺^{4}}{90} = {鈭�}_{n = 1}^{鈭�} \frac{1}{n^{4}}$

Thus $\frac{蟺^{4}}{90} = {鈭�}_{n = 1}^{鈭�} \frac{1}{n^{4}}$

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Most popular questions from this chapter

Represent each of the following functions (a) by a Fourier cosine integral; (b) by a Fourier sine integral. Hint: See the discussion just before Parseval鈥檚 theorem.

29. $f (x) = {\begin{cases} - 1, 0 < x < 2 \\ 1, 2 < x < 3 \\ 0, x > 3 \end{cases}$

Use the results ${鈭�}_{a}^{b} s i n^{2} k x d x = {鈭�}_{a}^{b} c o s^{2} k x d x = \frac{1}{2} (b - a)$ to evaluate the following integrals without calculation.

(a) ${鈭�}_{0}^{4 蟺 / 3} s i n^{2} (\frac{3 x}{2}) d x$

(b) ${鈭�}_{- 蟺 / 2}^{3 蟺 / 2} c o s^{2} (\frac{x}{2}) d x$

Find the fourier transform of $f (x) = e^{鈭� x^{2} / (2 蟽^{2})}$ . Hint: Complete the square in the xterms in the exponent and make the change of variable $y = x + 蟽^{2} i 伪$ .Use tables or computer to evaluate the definite integral.

Use Parseval鈥檚 theorem and the results of the indicated problems to find the sum of the series in Problems 5 to 9. The series ${鈭�}_{n = o d d}^{} \frac{1}{n^{4}}$ ,using problem 9.10.

Use Parseval鈥檚 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

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