# Poisson's fundamental theorem of calculus via Taylor's formula

## TL;DR

This paper revisits Poisson's proof of a version of the fundamental theorem of calculus using Taylor's formula with Lagrange remainder, focusing on Riemann sums with equally spaced endpoints, and discusses educational benefits.

## Contribution

It provides a modern adaptation of Poisson's proof employing Taylor's formula, offering insights for calculus education and alternative proof techniques.

## Key findings

- Offers a new proof approach using Taylor's formula
- Highlights educational benefits for calculus teaching
- Connects Riemann sums with classical calculus theorems

## Abstract

We use Taylor's formula with Lagrange remainder to make a modern adaptation of Poisson's proof of a version of the fundamental theorem of calculus in the case when the integral is defined by Euler sums, that is Riemann sums with left (or right) endpoints which are equally spaced. We discuss potential benefits for such an approach in basic calculus courses.