Please identify a series expansion

In summary, the conversation discusses a series expansion for a function f(x) and asks for help in identifying it. The expansion includes a constant and terms involving the difference Δ and its derivatives. It is believed that this result was obtained using Taylor's formula and is used in the Euler-Maclaurin summation formula. The speaker requests for the derivation of this result.
  • #1
Jaggis
36
0
Hi,

Could someone please identify the following series expansion for me, if possible:

$$f(x) = 1/h\int_x^{x+h}f(t)dt + A_{1}Δf(x) + A_{2}hΔf'(x) +...+ A_{m-1}h^{m-2}Δf^{(m-2)}(x) +r$$

where $$A_{m}$$ are, as far as I know, plain constants and $$Δ = [x, x+h]$$.

I think this result was partially obtained through using the Taylor's formula. It is a result that is used as a basis when constructing the Euler-Maclaurin summation formula.

I'd like to know how exactly this result follows. Thank you in advance.
 
Last edited:
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  • #2
Try looking for "Derivation of Euler-Maclaurin summation formula" - if you are correct you will find the related expressions.
 

Related to Please identify a series expansion

1. What is a series expansion?

A series expansion is a mathematical representation of a function as an infinite sum of terms. It is used to approximate a function and can provide more accurate values for a certain range of inputs.

2. Why is series expansion important?

Series expansion is important in many areas of science and engineering, as it allows for the approximation of complex functions and makes calculations easier. It also helps in understanding the behavior of functions and can be used to solve problems that cannot be solved using other methods.

3. How do you identify a series expansion?

To identify a series expansion, you need to look for a pattern in the terms of the function. The series expansion should have a common factor or term that is multiplied by a variable raised to different powers. You can also use mathematical formulas and techniques, such as the Taylor series or the Maclaurin series, to identify a series expansion.

4. Can a series expansion be used for any function?

No, a series expansion can only be used for functions that can be written as an infinite sum of terms. This means that the function must be continuous and differentiable in the interval where the expansion is being performed.

5. What are the limitations of using series expansion?

Series expansion can only provide an approximation of a function and is not an exact solution. The accuracy of the approximation also depends on the number of terms used in the expansion. Additionally, series expansion may not converge for certain functions or may require a large number of terms to converge, making it computationally expensive.

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