From some personal investigation, I've noticed that all convergence tests for infinite series (at least, the real kind) can be rephrased in terms of the discrete derivative $∆f(x)$ of a function $f(x)$, sometimes to give interesting results.
For example, the statement: $$\text{If } \lim_{k → ∞} a_k ≠ 0 \text{ then } ∑a_k \text{ diverges.}$$
Is equivalent to the statement: $$\text{If } \lim_{x → ∞} ∆f(x) ≠ 0 \text{ then } f(x) \text{ diverges.}$$ Except that the second statement is arguably more general, since $f(x)$ is not explicitly required to be in the form of a sum. Note that the limit of $∆ f(x)$ follows the definition of a limit of a sequence, rather than a function.
Another example, is the ratio test which is given: $$ \text{If } \lim_{k → ∞} \frac{a_{k+1}}{a_k} < 1 \text{ then } ∑a_k \text{ converges.}$$
This statement is equivalent to: $$ \text{If } \lim_{x → ∞} \frac{∆^{\!2} f(x)}{∆ f(x)} < 0 \text{ then } f(x) \text{ converges.}$$
Besides said generalization, this representation is also interesting because proving convergence tests in terms of $∆ f(x)$ and $f(x)$ does not require manipulating sums directly. Instead, the proofs can be given in terms of basic theorems about $∆ f(x)$. They generally resemble proofs you'd find in calculus.
Is anyone familiar with some thorough treatment of studying infinite series using discrete calculus in this way?
