Can a Bound be Found for the Error in Higher Order Taylor Series?

[hadron23]

Hello,

I am trying to come up with an expression for a bound on the sum of higher order terms, above second order. Consider the following Taylor expansion of a function f(x) around a point a,

$$f(x) = f(a) + \frac{f^{(1)}(a)}{1!}(x-a) + \frac{f^{(2)}(a)}{2!}(x-a)^2+ \frac{f^{(3)}(a)}{3!}(x-a)^3+...$$

Is it possible to come up with a value $$M$$ such that

$$\begin{align} &\left|\left|f(x) - [f(a) + \frac{f^{(1)}(a)}{1!}(x-a)]\right|\right|_2^2 \le M\\ &\Rightarrow \left|\left|\frac{f^{(2)}(a)}{2!}(x-a)^2+ \frac{f^{(3)}(a)}{3!}(x-a)^3+...\right|\right|_2^2 \le M \end{align}$$

That is, come up with an upper bound on the error of the higher order terms.

Thanks

 

[mathman]

I suspect you would need some conditions on the higher derivatives of f(x) at x=a. Otherwise they might get large fast.

 

[hgfalling]

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