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GATE 2022 ME (ME1) – Question 14

Engineering Mathematics · Calculus: Partial derivatives, Taylor series, maxima and minima, Fourier series · 1 mark · Multiple choice

The Fourier series expansion of $x^3$ in the interval $-1 \le x < 1$ with periodic continuation has

  1. only sine terms
  2. only cosine terms
  3. both sine and cosine terms
  4. only sine terms and a non-zero constant

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Show answer and explanation

Correct answer: (A) only sine terms

Explanation

The function $x^3$ is odd on the interval $[-1, 1)$, so all the cosine coefficients and the constant term (the mean) are zero. Only sine terms are left.