On the remainder in the Taylor theorem

TL;DR

This paper presents a simple, induction-based proof for the bound on the remainder term in Taylor's theorem, emphasizing accessibility for undergraduate students.

Contribution

It offers a straightforward proof of the Taylor remainder bound using only induction and monotonicity, making it suitable for educational purposes.

Findings

Provides a clear proof of the Taylor remainder bound

Uses only induction and monotonicity assumptions

Suitable for teaching undergraduate students

Abstract

We give a short straightforward proof for the bound of the reminder term in the Taylor theorem. The proof uses only induction and the fact that $f'\geq 0$ implies the monotonicity of $f$, so it might be an attractive proof to give to undergraduate students.