articles/discrete-calculus/discrete_calculus

[utterances-bot]

Discrete Calculus

https://mrmineev.com/articles/discrete-calculus/discrete_calculus.html

[browsersolutions]

while I was reading this I, kept thinking of Taylor series expansions. Could you elaborate on how a Taylor series differs from discrete calculus?

[MrMineev]

I may not fully understand your question, but here’s some clarification. Discrete calculus is a field, which deals with discrete functions, while Taylor series/expansion is a series used to approximate continuous functions using infinite polynomials. If you’re asking whether an analog of the Taylor series exists in discrete calculus, the answer is yes.

The discrete Taylor expansion is quite similar to the classical continuous version and is expressed as the following.

$$f(x) = \sum_{k=0}^\infty \frac{(x-x_0)_k}{k!} \Delta^k f(x_0)$$

Where $x_0$ is the base point of the expansion, just as in the classical case. This along with other similarities between discrete calculus and classical calculus should not be unexpected. Many proofs and ideas from classical calculus can often be adapted to discrete settings as well.