Question

Find the value of \(\frac{1}{1^2} +\frac{1}{3^2} +\frac{1}{5^2} +\frac{1}{7^2} \) +….when finding the Half range Fourier sine series of the function f(x) = 1 in 0<x<π.

(a) \(\frac{\pi^2}{4} \)

(b) \(\frac{\pi^2}{8} \)

(c) \(\frac{\pi^2}{2} \)

(d) \(3\frac{\pi^2}{8} \)

I had been asked this question in examination.

My question is based upon Fourier Half Range Series topic in chapter Fourier Series of Engineering Mathematics

Answer 1

Right choice is (b) \(\frac{\pi^2}{8} \)

To explain I would say: Using parseval’s relation for half range Fourier sine series,

\(\int_{-l}^l(f(x))^2 dx=l[\frac{a_0^2}{2}+∑_{n=1}^∞(a_n^2+b_n^2 ) ] \)

L.H.S. = \(\int_0^\pi(1)^2 dx = \pi \)

bn = \(\frac{2}{\pi} \int_0^π1.sin(nx)dx \)

= \(\frac{2}{\pi n} (1-(-1)^n ) \)

Using parseval’s formula,

\(\pi = \frac{\pi}{2} * \frac{4}{\pi^2} * 2 (\frac{1}{1^2} +\frac{1}{3^2} +\frac{1}{5^2} +\frac{1}{7^2} +….) \)

Therefore,

\(\frac{1}{1^2} +\frac{1}{3^2} +\frac{1}{5^2} +\frac{1}{7^2} +….= \frac{\pi^2}{8}.\)