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I have a problem to solve that consists in finding a frequency domain expression of this expression, the center of gravity of a sequence. I have tried in several manners but no sucess so far. Does anyone know a way and can give me some guidance?

Thank you all!enter image description here

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  • $\begingroup$ Hint: what is the derivative of the transformed function, evaluated at $0$. THis pulls down a factor of $n$ into each term. $\endgroup$ – Mark Fischler Apr 1 '15 at 23:47
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Let $$X(\omega) = \sum_n e^{i\omega n}x_n$$ Then $$X(0) = \sum_n x_n$$ and $$\left. \frac{dX(\omega)}{d\omega} \right|_{\omega = 0} = i\sum_n n x_n$$ So the answer is $$ -i \frac{\left. \frac{dX(\omega)}{d\omega} \right|_{\omega = 0}}{X(0)} $$

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