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Determine the Z-Transform for the following sequence:

$$ |n|(\frac{1}{2})^{|n|} $$

I have tried to solve the above problem. However, the answer that I got is the negative of what is given in the solution manual. What I may have done wrong?

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SOLUTION FROM SOLUTION MANUAL:

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  • $\begingroup$ What book is this from? Is this the official solution manual? $\endgroup$ – Matt L. May 19 '20 at 15:31
  • $\begingroup$ @MattL. This is from Oppenheim Signals and Systems. Yes, this is the official solution manual. $\endgroup$ – Soumee May 29 '20 at 13:08
  • $\begingroup$ Strange that there seem to be so many mistakes. $\endgroup$ – Matt L. May 29 '20 at 18:44
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It's always good to do a sanity check on such results. E.g., you could try to see that $X(1)$ equals the sum of all time domain samples:

$$X(1)=\sum_{n=-\infty}^{\infty}x[n]\tag{1}$$

With $x[n]=|n|\left(\frac12\right)^{|n|}$ it is clear that the result of $(1)$ must be positive. However, for the solution from the manual you get $X(1)<0$, whereas for your solution you obtain $X(1)>0$. So I think you can be confident that your solution is the correct one.

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