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What happens to the Fourier series representation if the period T of the function approaches infinity?
Question

What happens to the Fourier series representation if the period T of the function approaches infinity?

A.

Fourier series remains unchanged

B.

Fourier series becomes a Fourier transform

C.

Fourier series converts to a Laplace transform

D.

Fourier series diverges to infinity

Correct option is B

When the period T of a periodic function approaches infinity, the function effectively becomes aperiodic. In the Fourier series, frequency components are discrete, spaced by:Δω=2πT As T, Δω0Thus, the discrete spectrum becomes continuous, and the Fourier series representation transitions into the Fourier transform.\text{When the period } T \text{ of a periodic function approaches infinity, the function effectively becomes aperiodic.} \\[8pt]\bullet \ \text{In the Fourier series, frequency components are discrete, spaced by:} \\[6pt]\Delta \omega = \frac{2\pi}{T} \\[8pt]\bullet \ \text{As } T \rightarrow \infty, \ \Delta \omega \rightarrow 0 \\[10pt]\text{Thus, the discrete spectrum becomes continuous, and the Fourier series representation transitions into the Fourier transform.}​​

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