Question:
The Fourier series expansion of \( x^3 \) in the interval \( -1 \leq x \leq 1 \) with periodic continuation has
The Fourier series expansion of \( x^3 \) in the interval \( -1 \leq x \leq 1 \) with periodic continuation has
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For odd functions, the Fourier series consists of only sine terms, since sine functions are odd.
Updated On: Dec 15, 2025
- only sine terms
- only cosine terms
- both sine and cosine terms
- only sine terms and a non-zero constant
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The Correct Option is A
Solution and Explanation
Step 1: Analyze the function \( x^3 \).
The function \( x^3 \) is an odd function, meaning \( f(-x) = -f(x) \). The Fourier series expansion of an odd function contains only sine terms, as sine functions are odd. Step 2: Conclusion.
Thus, the Fourier series expansion of \( x^3 \) will have only sine terms. Final Answer: \[ \boxed{\text{Only sine terms}} \]
The function \( x^3 \) is an odd function, meaning \( f(-x) = -f(x) \). The Fourier series expansion of an odd function contains only sine terms, as sine functions are odd. Step 2: Conclusion.
Thus, the Fourier series expansion of \( x^3 \) will have only sine terms. Final Answer: \[ \boxed{\text{Only sine terms}} \]
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