# Determine the Z-Transform for the following sequence: $|n|(\frac{1}{2})^{|n|}$

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?

SOLUTION FROM SOLUTION MANUAL:

• What book is this from? Is this the official solution manual? – Matt L. May 19 '20 at 15:31
• @MattL. This is from Oppenheim Signals and Systems. Yes, this is the official solution manual. – Soumee May 29 '20 at 13:08
• Strange that there seem to be so many mistakes. – Matt L. May 29 '20 at 18:44

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.