Delta Function and Unit Step Formula

Question

I have come across the following formula:

$$u(n)=\sum_{m=-\infty}^{n}\delta(m)$$

where $u(n)$ is the Unit Step and $\delta(m)$ is the Delta Function:

What I can't understand is how this formula "works".

Expanding the formula we have:

$$u(n)=...+\delta(0)+\delta(1)+\delta(2)+...+\delta(n-1)+\delta(n)=\delta(0)=1$$

So expanding it, no matter what, gives us the same result which obviously is not a Unit Step, so I can't understand how that formula can produce the Unit Step.

I know that I have made a mistake somewhere but I don't know where. Can someone explain to me that formula and my mistake?

Answer 1

If $n<0$, then $\delta(m) = 0$ for all $m \le n$, and so $u(n) = 0$.

if $n \ge 0$, then $\delta(0) = 0$ and $\delta(m) = 0$ for all $n \neq 0$, hence $u(n) = 1$.

Answer 2

This is because $u(n)=1$ if $n \geq 0$ and $u(n)=0$ if $n \leq 0$. In this last cases, $\delta (0)$ never appears, so $u(n)=0$.