What is the Generalised Mean Value Theorem?

[pivoxa15]

Does anyone know the mean value theorem associated with the Taylor series. Representing the Taylor series a finite sum and an end term? I don't get how they get it to look that way.

 

[quasar987]

can you be more specific about what you'Re asking?

 

[quetzalcoatl9]

i have seen the taylor series derived that way..you consider the n-th antiderivative of the n derivative. when you apply the fundamental thm of calculus you will get a series of terms that define the taylor series, with a remainder term. you use the mean value theorem to show that the remainder will go to 0 as long as your function grows no faster than n!

 

[pivoxa15]

i have seen the taylor series derived that way..you consider the n-th antiderivative of the n derivative. when you apply the fundamental thm of calculus you will get a series of terms that define the taylor series, with a remainder term. you use the mean value theorem to show that the remainder will go to 0 as long as your function grows no faster than n!

This could be it but I don't follow it very well.

 

[quetzalcoatl9]

This could be it but I don't follow it very well.

i know, but i don't feel like typing out all of the latex. just write out taking N definite integrals over some interval for a function that has been differentiated N times.

Arfken does exactly this derivation in his book (pg. 260 in the 2nd edition)

 

[jostpuur]

http://en.wikipedia.org/wiki/Taylor's_theorem

Quite good explanations. This stuff was left unclear to me when I was sleeping in the calculus classes, but now I feel like understanding the integral trickery that is done in the Wikipedia's article.

 

[quetzalcoatl9]

http://en.wikipedia.org/wiki/Taylor's_theorem

Quite good explanations. This stuff was left unclear to me when I was sleeping in the calculus classes, but now I feel like understanding the integral trickery that is done in the Wikipedia's article.

that is exactly the method that i was referring to. as long as the function doesn't grow faster than n! you will be fine and the remainder term will go to zero

<happy 600 posts to me>

 

[pivoxa15]

http://en.wikipedia.org/wiki/Taylor's_theorem

Quite good explanations. This stuff was left unclear to me when I was sleeping in the calculus classes, but now I feel like understanding the integral trickery that is done in the Wikipedia's article.

THe problem is they still didn't mention how the mean value theorem works in the proof.

 

[quetzalcoatl9]

THe problem is they still didn't mention how the mean value theorem works in the proof.

you invoke the mean value thm in order to convert the integral into the remainder term. in other words,

$$\int_a^b f(x) dx = (b - a)f(y)$$

where y is an element of the interval (a,b)

 

[pivoxa15]

I've seen a proof of the genearlised mean value theorem in books and its long but it finally explained it fully in terms of Taylor series.

 

[quetzalcoatl9]

i seem to recall that the MVT can be proven all by itself, and that the fundamental theorem of calculus can be derived from it

 

[pivoxa15]

But the generalised MVT is something else. It involves using induction.