Invariance of $T[x[n]]=\sum_{k=n-1}^{n+2}x[k]$ I have to test whether the following system is invariant or not: $T[x[n]]=\sum_{k=n-1}^{n+2}x[k]$, so I want to verify that, if $y[n]=T[x[n]]$, then $y[n-N]=T[x[n-N]]$.
$$T[x[n-N]]=\sum_{k=n-1}^{n+2}x[k-N]$$
$$y[n-N]=\sum_{k=n-N-1}^{n-N+2}x[k]$$
In the first equation, if $k'=k-N$, we have that $T[x[n-N]]=\sum_{k'=n-N-1}^{n-N+2}x[k']$, so both sums are equal and thus the system is invariant. Is this correct or am I making wrong assumptions? Thanks in advance!
Yes your computation is correct. Indeed, you could have seen this even easier, had you considered that the output computation sum was

$$ y[n] = T\{x[n]\} = x[n-1] + x[n] + x[n+1] + x[n+2] .$$

And this is clearly an LTI (linear time-invariant), FIR (finite impulse response) system with associated impulse response 

$$h[n] = \delta[n-1] + \delta[n] + \delta[n+1] + \delta[n+2] $$