Discrete Analogue of the Fundamental Theorem of Calculus

The first equation, when the operator is written inside the sum is meant to be analogous to $$\int^b_a f'(x) dx = f(b) - f(a).$$

In the second equation when the operator is written outside the sum, we are reminded of: $$ \frac{d}{dx} \int^x_a f(t) dt = f(x).$$

While the forward difference operator is indeed linear, we can not simply move the operator inside the summation. This is because in the second equation, the operator is acting on the index $n$, not on $k$ as it does in the first equation. We are considering the sequence $a_n = \displaystyle \sum_{k=n_0}^{n-1} x(k)$ so $$ \begin{equation} \Delta \left(\sum_{k=n_0}^{n-1}x(k) \right) = \Delta (a_n) = a_{n+1} - a_n = \sum_{k=n_0}^{n} x(k) -\sum_{k=n_0}^{n-1} x(k)= x(n) \end{equation} $$

so the second equation is correct as written.