Time invariance of a summation sequence
Sequences that are &quot;time-invariant&quot; must be constants (not varying with time at all).
What the OP is actually asking about is a (discrete-time) system whose response $y$ to the input $x$ is defined to be
$$y[n] = \begin{cases}\sum_{k=n_0}^n x[k], &amp; n \geq n_0,\\??? &amp; n &lt; n_0\end{cases}$$
where the $???$ is there because according to some traditionalists, if $n &lt; n_0$, then that sum is an empty sum which has value $0$ but more liberal-minded might treat that sum as representing $$x[n] + x[n+1] + x[n+2] + \cdots + x[n_0-1] + x[n_0]$$ (with more serious-minded folks even negating that sum in analogy with the standard result that for $a&lt;b$, $\int_b^a = -\int_a^b$). So, for the traditional meaning of things and $n_0$ denoting a (finite) integer, the system is not a time-invariant system. It is time-invariant (with unit pulse response $h[n]$ being the unit step function $u[n]$) if $n_0 = -\infty$ as noted in a comment by Robert Bristow-Johnson on the main question.