How to Differentiate and Integrate Sequences

The main purpose of this note is to show how to apply "discrete calculus" to solve certain discrete differential equations. Along the way we easily obtain closed formulas for sums such as E>L3 km, and prove discrete analogues of Taylor's formula and the Chain Rule (involving moving averages). Discrete integrating factors allow us to express the solutions of certain of our discrete differential equations naturally as convolutions. Such a solution may be interpreted as a convolution of the "impulse response" of the system with the "driving sequence". This has an analog in differential equations: The driven harmonic oscillator whose solution is the convolution of the "impulse response" with the driver. Our main point is to use one's calculus/differential equations intuition to solve certain discrete differential equations, whose solutions are usually found by guessing and induction. So let us begin by first discussing discrete calculus. All of our sequences are sequences of real or complex numbers