Q4Advanced Engineering Mathematics

To determine the inverse Z-transform of $\log \left(\frac{z}{z+1}\right)$, we algebraically rewrite the expression in the form $-\log (1+z^{-1})$. We then utilize the standard infinite Taylor series expansion for the natural logarithm $\ln (1+x)$. By directly comparing the resulting coefficients of the $z^{-n}$ terms with the standard definition $U(z)=\sum u_n{z}^{-n}$, we successfully identify the discrete sequence values as $u_n=\frac{(-1{)}^n}{n}$ for all $n>0$, and zero otherwise.