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Chapter 7: Q7P (page 378)

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=1鈭�1n2,using problem 5.8.

Short Answer

Expert verified

By Parseval theorem蟺26=鈭�n=1鈭�1n2

Step by step solution

01

Given Information.

The given series is鈭�n=1鈭�1n2.The sum of the series is to be found out.

02

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

03

Sum of the series

It is know that the solution of the problem 5.8 isf(x)=1-2鈭�n=1鈭�sinnxn-1n

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

<f(x)2>=12蟺鈭�-蟺蟺1+x2dx=12蟺1+x3|蟺-蟺=1+蟺23

Use the Parseval theorem

1+蟺23=1+12鈭�n=1鈭�2-1nn2鈬�蟺26=鈭�n=1鈭�1n2

Thus 蟺26=鈭�n=1鈭�1n2

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