Determine the Z-Transform for the following sequence: $ |n|(\frac{1}{2})^{|n|} $
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 $X(1)&lt;0$, whereas for your solution you obtain $X(1)&gt;0$. So I think you can be confident that your solution is the correct one.