LTI-properties of a system summing infinite number of input values

A pop-quiz featured a system analysis of the following system: $$y[n]=\sum_{n=0} ^{-\infty} x[n]$$

The professor argued that this system exhibits the following properties:

• Linearity
• Causality
• Instability
• Memory
• Time-invariance

In general the system seems nonsensical, as it sums all samples from the beginning of time up until now, even though these are not accessible to $$x[n]$$. Colleagues argued that the index $$n$$ might be "local" to the sum, translating to a more sensical description $$y[n]=\sum_{k=0} ^{-\infty} x[n+k]$$ with a distinguished index $$k$$.

EDIT:
As provided by the answer below, the most likely meant system is $$y[n]=\sum_{k=-\infty} ^{n} x[k]$$

Note:
We do not intend to use dsp.stackexchange as a lazy solution but as a source of discussion, as this topic is of personal curiosity.

• If you're doing theoretical analysis, it's fair to claim knowledge of $x[n]\ \forall \ n \in (-\infty, 0]$. That's not the problem. However, if you look at the expression given, you'll see that it sums from $k = -\infty$ up to $k = 0$ (variable change mine) regardless of the current time. So it's just a constant. Oct 31, 2022 at 20:55
• For that matter, for really theoretical analysis, you can claim knowledge of $x[n]$ for all time -- it really depends on the nature of the analysis you're doing. E.g., Fourier transforms assume that a signal is known for all time, even though that's physically impossible. Nov 1, 2022 at 15:08

The given equation relating the output to the input doesn't make sense. Note that on the left-hand-side the variable $$n$$ is used as time (or anything else) index, but on the right-hand-side it is used as the summation index. What is probably meant is the following system:

$$y[n]=\sum_{k=-\infty}^nx[k]\tag{1}$$

which is just an accumulator, and which has all the properties listed in the question.

Note that $$(1)$$ can be implemented recursively:

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

The system's impulse response is the unit step sequence: $$h[n]=u[n]$$. It is straightforward to show that $$(1)$$ is equivalent to

$$y[n]=(x\star h)[n]=\sum_{k=-\infty}^{\infty}x[k]h[n-k]=\sum_{k=-\infty}^{\infty}x[k]u[n-k]\tag{3}$$