Estimates of the remainder in Taylor’s theorem using the Henstock–Kurzweil integral

When a real-valued function of one variable is approximated by its n th degree Taylor polynomial, the remainder is estimated using the Alexiewicz and Lebesgue p-norms in cases where f (n) or f (n+1) are Henstock–Kurzweil integrable. When the only assumption is that f (n) is Henstock–Kurzweil integrable then a modified form of the n th degree Taylor polynomial is used. When the only assumption is that f (n) ∈ C 0 then the remainder is estimated by applying the Alexiewicz norm to Schwartz distributions of order 1.