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I determined the complex Fourier series of a sinusoidal signal and arrived at the following expression: $$\sum_{n=\infty}^{\infty} \left[\frac{4e^{-j \frac{\pi}{2}n}}{\pi(1-n^2)}(e^{-j\pi n}+1)\right]e^{-j \frac{\pi}{2}tn} $$

However, we have a division by zero at $n = 1$ and $n = -1 $. Is there any way to determine $D_{-1}$ and $D_1$?

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  • $\begingroup$ You don't have a division by $0$ at $n=\pm 1$, you have the indeterminate form $\dfrac 00$ because $e^{-jn\pi}+1 = 0$ when $n=\pm 1$. You could use L'Hopital's rule (treating $n$ as a continuous variable instead of being integer-valued), but it is better to follow MattL's suggestion and evaluate separately for $n=\pm 1$. $\endgroup$ Commented Mar 26, 2021 at 14:26

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HINT:

You need to go back to the original definition of the Fourier coefficients via the integral, and treat $|n|=1$ as a special case. Your mistake happened there already (you divided by zero).

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  • $\begingroup$ Can I just calculate the integral for $n = 1$ and $n = -1$ to determine these coefficients? $\endgroup$
    – carraro
    Commented Mar 26, 2021 at 12:40
  • $\begingroup$ @JulyH.: Yes, sure, why not? The definition is valid for any value of $n$. $\endgroup$
    – Matt L.
    Commented Mar 26, 2021 at 13:47

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