Geodesic
Estimates of the remainder in Taylor’s theorem using the Henstock-Kurzweil integral
Czechoslovak Mathematical Journal, Tome 55 (2005) no. 4, pp. 933-940

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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 $f^{(n)}\in C^0$ then the remainder is estimated by applying the Alexiewicz norm to Schwartz distributions of order 1.
Classification : 26A24, 26A39
Keywords: Taylor’s theorem; Henstock-Kurzweil integral; Alexiewicz norm 
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Talvila, Erik. Estimates of the remainder in Taylor’s theorem using the Henstock-Kurzweil integral. Czechoslovak Mathematical Journal, Tome 55 (2005) no. 4, pp. 933-940. http://geodesic.mathdoc.fr/item/CMJ_2005__55_4_a9/