Question

Consider the discrete time signal

$x\left(n\right)={\left(\frac{1}{3}\right)}^{n}u\left(n\right),whereu\left(n\right)=\{\begin{array}{cc}1,& n\ge 0\\ 0,& n<0\end{array}$
and $y\left(n\right)=x(-n),-\infty <n<\infty$

Then $\sum _{n=-\infty }^{\infty }y\left(n\right)$ is______.

• A. 1.5

Answer 1

The correct option is A 1.5
Given, $x\left(n\right)={\left(\frac{1}{3}\right)}^{n}u\left(n\right):n\ge 0$

$y\left(n\right)=x(-n)={\left(\frac{1}{3}\right)}^{-n}u(-n).n\le 0$

$\sum _{n=-\infty }^{\infty }y\left(n\right)$=$\sum _{n=-\infty }^{\infty }{\left(\frac{1}{3}\right)}^{-n}u(-n)$

$=\sum _{n=-\infty }^0{\left(\frac{1}{3}\right)}^{-n}u(-n)=\frac{1}{1-\left(\frac{1}{3}\right)}=\frac{1}{2/3}=\frac{3}{2}=1.5$