The Maclaurin Expansions

The papers [15], [16], [4], [12], [2], [14], [5], [1], [3], [7], [6], [10], [11], [8], [9], [17], and [13] provide the notation and terminology for this paper. The following proposition is true (1) For every real number x and for every natural number n holds |x| = |x|. Let f be a partial function from R to R, let Z be a subset of R, and let a be a real number. The functor Maclaurin(f, Z, a) yields a sequence of real numbers and is defined by: (Def. 1) Maclaurin(f, Z, a) = Taylor(f, Z, 0, a). The following propositions are true: (2) Let n be a natural number, f be a partial function from R to R, and r be a real number. Suppose 0 < r and f is differentiable n + 1 times on ]−r, r[. Let x be a real number. Suppose x ∈ ]−r, r[. Then there exists a real number s such that 0 < s and s < 1 and f(x) = ( ∑κ α=0(Maclaurin(f, ]−r, r[, x))(α))κ∈N(n) + f (]−r,r[)(n+1)(s·x)·x (n+1)! . (3) Let n be a natural number, f be a partial function from R to R, and x0, r be real numbers. Suppose 0 < r and f is differentiable n + 1 times on ]x0 − r, x0 + r[. Let x be a real number. Suppose x ∈ ]x0 − r, x0 + r[. Then there exists a real number s such that 0 < s and s < 1 and |f(x) − ( ∑κ α=0(Taylor(f, ]x0 − r, x0 + r[, x0, x))(α))κ∈N(n)| = | (]x0−r,x0+r[)(n+1)(x0+s·(x−x0))·(x−x0) (n+1)! |.