# Indetermination in a complex fourier series

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 $$D_1$$?

• 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 $n=\pm 1$. Mar 26, 2021 at 14:26

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