I came across a question while studying signals and systems, and I am not able to prove it. The question wants me to show that the following equation is shift invariant and linear. Showing that it is linear is not too difficult, but I am having a hard time showing that it is shift invariant.
$y(n)=\sum_{m=-\infty}^{n} x(m)$
Thank you.
If the input $x(n)$ is shifted by $k$ units, we have $$ y(n,k)=\sum^n_{m=-\infty}x(m-k) $$ From the definition of shift invariant systems, we must have $$ y(n-k)=\sum^{n-k}_{m=-\infty}x(m)=\sum^{n}_{v=-\infty}x(v-k) $$
The right hand sides are equal therefore the system is shift invariant.