The Simplest Proof of Taylor's Theorem: An Easy Guide

Taylor's Theorem is a foundational concept in mathematics that allows us to approximate complex functions using polynomials. This theorem is particularly useful in various fields such as calculus, engineering, and physics, where understanding the local behavior of a function is crucial.

What is Taylor's Theorem?

Taylor's Theorem provides a powerful method to approximate a function using polynomials based on its derivatives at a specific point. The simplest form of Taylor's Theorem states that if a function f is n-times differentiable at a point a, it can be expressed as:

fx P_nx R_nx

where P_nx is the n-th degree Taylor polynomial given by:

P_nx fa f'x - a frac{f''(a)(x - a)^2}{2!} cdots frac{f^{(n)}(a)(x - a)^n}{n!} R_nx

and R_nx is the remainder term that captures the error in the approximation.

Proof Outline

Let's break down the proof of this theorem into several steps:

Define the Remainder

The remainder R_nx can be defined in various forms, one of which is the Lagrange form of the remainder:

R_nx frac{f^{(n-1)}(c)(x - a)^{n-1}}{(n-1)!}

for some c between a and x.

Construct the Function

Define a new function g(t) as:

g(t) f(a) f'(a)t frac{f''(a)t^2}{2!} cdots frac{f^{(n)}(a)t^n}{n!}

This function is differentiable in the interval [0, x - a].

Apply the Mean Value Theorem

By the Mean Value Theorem, we can express g(t) in terms of its derivatives at t 0:

g(t) g(0) g'(0)t frac{g''(0)t^2}{2!} cdots frac{g^{(n)}(0)t^n}{n!} R_n(t)

Substitute t x - a to get:

fx fa f'x - a frac{f''(a)(x - a)^2}{2!} cdots frac{f^{(n)}(a)(x - a)^n}{n!} R_nx - a

This shows that fx can be expressed as the sum of the Taylor polynomial P_nx and the remainder term R_nx.

Summary

This proof outlines the basic structure of Taylor's Theorem using the definition of the remainder and the application of the Mean Value Theorem. The choice of the remainder form like Lagrange's form helps in understanding the accuracy of the polynomial approximation.

Understanding Taylor's Theorem is not just about the mathematical derivation; it is about the intuition behind it. It allows us to simplify complicated functions around a point, making analysis and problem-solving more manageable in various scientific and engineering applications.