The Fourier series expansion of the function $ f(x) = |\cos(x)| $ in the interval $ (-\pi, \pi) $ is $ \alpha + \beta \left[ \dfrac{1}{3}\cos(2x) - \dfrac{1}{15}\cos(4x) + \cdots \right] $.
The values of $ \alpha $ and $ \beta $ respectively are ______.

Question

Let $ g(x) = x^2 $, $ -\pi \le x \le \pi $. The coefficient of $ \cos(3x) $ in the Fourier series expansion of $g(x)$ is:

Answer 1

The Fourier expansion of $|\cos x|$ on $(-\pi,\pi)$ is a standard result that can be quoted and then matched term by term to the series given in the question.

State the known expansion: $|\cos x| = \dfrac{2}{\pi} + \dfrac{4}{\pi}\left[\dfrac{1}{1 \cdot 3}\cos(2x) - \dfrac{1}{3 \cdot 5}\cos(4x) + \dfrac{1}{5 \cdot 7}\cos(6x) - \cdots \right]$, obtained from the general formula $|\cos x| = \dfrac{2}{\pi} + \dfrac{4}{\pi}\sum_{n=1}^{\infty} \dfrac{(-1)^{n+1}}{4n^2-1}\cos(2nx)$.
Simplify the first two bracket terms: $\dfrac{1}{1 \cdot 3} = \dfrac{1}{3}$ and $\dfrac{1}{3 \cdot 5} = \dfrac{1}{15}$, so the bracket reads $\dfrac{1}{3}\cos(2x) - \dfrac{1}{15}\cos(4x) + \cdots$, which is precisely the bracket printed in the question.
Match the outside factors: Comparing $\dfrac{2}{\pi} + \dfrac{4}{\pi}\left[\cdots\right]$ with $\alpha + \beta\left[\cdots\right]$ term by term gives $\alpha = \dfrac{2}{\pi}$ and $\beta = \dfrac{4}{\pi}$ directly, with no extra computation needed once the bracket is checked to line up.

So $\alpha = 2/\pi$ and $\beta = 4/\pi$, which is option A.

Answer 2

The Fourier expansion of $|\cos x|$ on $(-\pi,\pi)$ is a standard result that can be quoted and then matched term by term to the series given in the question.

State the known expansion: $|\cos x| = \dfrac{2}{\pi} + \dfrac{4}{\pi}\left[\dfrac{1}{1 \cdot 3}\cos(2x) - \dfrac{1}{3 \cdot 5}\cos(4x) + \dfrac{1}{5 \cdot 7}\cos(6x) - \cdots \right]$, obtained from the general formula $|\cos x| = \dfrac{2}{\pi} + \dfrac{4}{\pi}\sum_{n=1}^{\infty} \dfrac{(-1)^{n+1}}{4n^2-1}\cos(2nx)$.
Simplify the first two bracket terms: $\dfrac{1}{1 \cdot 3} = \dfrac{1}{3}$ and $\dfrac{1}{3 \cdot 5} = \dfrac{1}{15}$, so the bracket reads $\dfrac{1}{3}\cos(2x) - \dfrac{1}{15}\cos(4x) + \cdots$, which is precisely the bracket printed in the question.
Match the outside factors: Comparing $\dfrac{2}{\pi} + \dfrac{4}{\pi}\left[\cdots\right]$ with $\alpha + \beta\left[\cdots\right]$ term by term gives $\alpha = \dfrac{2}{\pi}$ and $\beta = \dfrac{4}{\pi}$ directly, with no extra computation needed once the bracket is checked to line up.

So $\alpha = 2/\pi$ and $\beta = 4/\pi$, which is option A.

The Fourier series expansion of the function \( f(x) = |\cos(x)| \) in the interval \( (-\pi, \pi) \) is \( \alpha + \beta \left[ \dfrac{1}{3}\cos(2x) - \dfrac{1}{15}\cos(4x) + \cdots \right] \).
The values of \( \alpha \) and \( \beta \) respectively are ______.

Show Hint

The Correct Option is A

Solution and Explanation

Top Questions on Fourier series

Questions Asked in GATE NM exam

The Fourier series expansion of the function \( f(x) = |\cos(x)| \) in the interval \( (-\pi, \pi) \) is \( \alpha + \beta \left[ \dfrac{1}{3}\cos(2x) - \dfrac{1}{15}\cos(4x) + \cdots \right] \). The values of \( \alpha \) and \( \beta \) respectively are ______.