GATE 2020 CE (CE2) – Question 49
For $x-x^2$ on $-\pi\le x\le\pi$, Fourier expansion is $$x-x^2=\frac{a_0}{2}+\sum_{n=1}^{\infty}a_n\cos nx+\sum_{n=1}^{\infty}b_n\sin nx.$$ Find $a_0$ to two decimals.
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Correct answer: -6.61 to -6.55
Explanation
The constant coefficient is $a_0=\pi^{-1}\int_{-\pi}^{\pi}(x-x^2)\,dx$. The odd term integrates to zero. Thus $a_0=-2\pi^2/3=-6.579736$, or **−6.58**.