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GATE 2022 CE (CE1) – Question 30

Engineering Mathematics · PDE: Fourier series, separation of variables, diffusion, wave and Laplace equations · 1 mark · Numerical answer

For $f(x)=\sum_{n=0}^{\infty}f_n\cos nx=\cos^4x$, find $f_4+f_5$ (three decimals).

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Correct answer: 0.115 to 0.135

Explanation

Twice apply the double-angle identity to obtain $\cos^4x=(3+4\cos2x+\cos4x)/8$.

Therefore the cosine-series coefficients are $f_4=1/8$ and $f_5=0$. Their sum is **0.125**.