The Taylor Expansions

For simplicity, we use the following convention: n denotes a natural number, i denotes an integer, p, x, x0, y denote real numbers, q denotes a rational number, and f denotes a partial function from R to R. Let q be an integer. The functor qZ yields a function from R into R and is defined as follows: (Def. 1) For every real number x holds (qZ)(x) = x q Z. Next we state a number of propositions: (1) For all natural numbers m, n holds x Z = (x n Z) · x m Z . (2) Z is differentiable in x and ( n Z) (x) = n · x Z . (3) If f is differentiable in x0, then ( n Z) · f is differentiable in x0 and (( n Z) · f)′(x0) = n · f(x0) n−1 Z · f ′(x0). (4) exp(−x) = 1 expx .