Stability of the summation function Stability of the below function was asked.

$$
T(x[n]) = \sum_{k=n_0}^n x[n]
$$

I understand that $x[n]$ shouldn't exceed $M$ value. $x[n]\le M$. I mean, we are looking for if the input is bounded, what will the output be.

However, i didn't understand from where $|n-n_0|M$ came from in the below function.

$$
\left |T(x[n]) \right| \le \sum_{k=n_0}^n \left| x[n] \right| \le |n-n_0|M
$$
Well, we are summing over $k$ in the range $n_0$ to $n$. The number of elements in the sum is therefore $n-n_0$. We also know that the sum must be positive and that all terms in the sum ($x[n]$) have absolute value less than $M$.

To account for $n-n_0$ being negative, but our sum MUST be positive, we take its absolute value.