Proving Lagrange's Remainder from Taylor's Theorem

[vanmaiden]

Homework Statement

I'm interested in the use of the mean value theorem of integration to convert Taylor's theorem into Lagrange's remainder. Though, I'm confused as to how it's used in the conversion. Could someone explain to me how the MVT of integration is used to convert Taylor's theorem into Lagrange's remainder?

Homework Equations

I"m bad with LaTeX, but it's Taylor's theorem right here: http://upload.wikimedia.org/math/6/d/7/6d70410a12402233d48598420cb7995f.png
and Lagrange's remainder right here: http://upload.wikimedia.org/math/2/1/2/212f9b5c042521027f93d959faf79798.png

The Attempt at a Solution

I reviewed the mean value theorem for integration, but I'm rather clueless as to how it's used to convert Taylor's theorem into Lagrange's remainder.

 

[mighty2000]

Can someone please provide a clear explanation?

Dear forum post author,

Thank you for your interest in understanding the connection between the mean value theorem of integration and the conversion of Taylor's theorem into Lagrange's remainder. Let me explain it in simple terms for you.

First, let's review Taylor's theorem. This theorem states that any function f(x) can be approximated by its Taylor series expansion at a given point a. This expansion includes the function value and its derivatives at that point a. However, this approximation may not be accurate for all values of x, especially for large values. This is where the Lagrange's remainder comes in.

Lagrange's remainder is a way to estimate the error between the actual function and its Taylor series approximation at a given point. It uses the mean value theorem of integration to do so. The mean value theorem of integration states that for a continuous function f(x) on the interval [a,b], there exists a point c between a and b where the integral of f(x) over [a,b] is equal to the product of the function value at c and the length of the interval (b-a).

Now, let's apply this to Taylor's theorem. If we consider the integral of the remainder term in Taylor's theorem over the interval [a,x], where x>a, we can use the mean value theorem of integration to find a point c between a and x where the integral is equal to the product of the function value and the length of the interval (x-a). This point c is also known as the Lagrange's point.

Thus, we can rewrite the remainder term in Taylor's theorem as the product of the function value at Lagrange's point and the remaining terms in the Taylor series expansion. This is known as the Lagrange's remainder form of Taylor's theorem.

I hope this explanation helps you understand the connection between the mean value theorem of integration and the conversion of Taylor's theorem into Lagrange's remainder. Keep exploring and asking questions to deepen your understanding of these concepts. Best of luck in your studies!