Newton's divided difference interpolation formula is the analogue of the Gregory-Newton and Taylor series for divided differences.

If $f$ is a real function and $x_0, x_1, \ldots$ is a sequence of distinct real numbers, then we have, for any integer $n > 0$ $$ f (x) = f(x_0) + (x - x_0) \Delta f (x_0, x_1) + \cdots + (x - x_0) \cdots (x - x_{n-1}) \Delta^n f (x_0, \ldots x_n) + R$$ where the remainder can be expressed either as $$R = (x - x_0) \cdots (x - x_n) \Delta^{n+1} f (x, x_1, \ldots, x_n)$$ or as $$R = {1 \over (n+1)!} (x - x_0) \cdots (x - x_n) f^{(n+1)} (\eta)$$ where $\eta$ lies between the smallest and the largest of $x, x_0, \ldots, x_n$

Remark. If $f$ is a polynomial of degree $n$ then $R$ vanishes.