# Issue evaluating the Fourier Transform of a box function at $w =0$

In my class we derived that the Fourier Transform of $$X[n] = u[n] - u[n-7]$$ is

$$\sum_{n=-\infty}^{\infty} X[n] e^{-jwn} = \sum_{n=0}^{6}e^{-jwn}$$ using the finite sum formula we get:

$$\frac{1-e^{-jw7}}{1-e^{-jw}} =\frac{e^{-jw7/2}(e^{-jw7/2}-e^{-jw7/2})}{e^{-jw/2}(e^{-jw/2}-e^{-jw/2})} = e^{-j3w} \frac{\sin(w7/2)}{\sin(w/2)}$$

if we evaluate $$e^{-j3w} \frac{\sin(w7/2)}{\sin(w/2)}$$ at $$w =0$$we get $$\frac{0}{0}$$. However if we plug in $$w = 0$$ into $$\sum_{n=-\infty}^{\infty}X[n]e^{-j0n} = 0$$ . Since the equality sign was used thrroughout all this, I am confused how one expression evaluates to a number while the other does not. Is there an underlying assumption of using the finite sum formula which causes this discrepancy ?

The finite geometric sum formula requires that your ratio not be equal to one, so you will have to consider the cases $$w \ne 0$$ and $$w = 0$$ separately. (Luckily, in the case of the ratio being equal to $$1$$, it is very trivial to get the end result.)