Notes on reading p. 39 of J. Hellerstein, et. al.’s Feedback Control of Computing Systems ¹ about “Linear, Time-Invariant Different Equations”.

\[ y(k+1) = \sum_{i=0}^k u(i)\]

\[ y(k) = \sum_{i=0}^{k-1} u(i)\]

\[ y(k + 1) = u(k) + \sum_{i=0}^{k-1} u(i)\]

With substitution we can get to Equation 2.3 in the book:

\[ y(k + 1) = u(k) + y(k)\]