# Discrete time fourier transform of partial sum

I came across the following property of the DTFT:

$\mathcal{F} \Bigg(\sum_{m=- \infty}^{n}x[m]\Bigg) = \frac{1}{1- e^{-j \omega}} X(e^{-j \omega}) + \pi X(e^{-j0}) \sum_{m= -\infty}^{\infty}\delta({\omega- 2\pi k)}$

where capital X denotes the DTFT of the sequence x. I would really like to see how this is proved in a formal way as I cannot get there myself. Can anyone help me?

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I don't see how this can be correct as the right hand side does not depend on $n$ in any way. Since the Fourier transform is isomorphic, $\mathcal{F}\left( \sum_{m=-\infty}^{n} x[m] \right) \neq \mathcal{F}\left( \sum_{m=-\infty}^{n+1} x[m] \right)$ if $x[n+1] \neq 0$. I think you need to double check the property. –  AnonSubmitter85 Jul 29 '13 at 21:19