Geometric intepretation of Taylor series

In summary, the conversation discusses the geometric interpretation of moments, which is defined as the integral of x^p multiplied by the probability density function. This can be compared to the computation of Fourier series using inner products with orthogonal basis functions. However, the basis polynomials used in moments are not always orthogonal, leading to the question of their geometric meaning and relationship to Taylor series.
  • #1
mnb96
715
5
Sorry, the title should be: geometric intepretation of moments

My question is:
does the formula of the moments have a geometrical interpreation?
It is defined as: [tex]m(p) = \int{x^{p}f(x)dx}[/tex]
If you can't see the formula it is here too: http://en.wikipedia.org/wiki/Moment_(mathematics) with c=0.

For example the Fourier series, are computed by inner products of the original function with all the basis functions (which are orthogonal): this means we are essentially finding the projections of the function onto the basis.

For the moments formula, we are computer inner products between the function and the "basis" polynomials 1, x, x^2, x^3, ... which are not always orthogonal.
What's the geometrical meaning of this? if any?
 
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  • #2
What does this have to do with Taylor series?
 
  • #3
Nothing indeed!
I was a bit distracted while I was writing the title, and then I was unable to correct it.
Apparently you were no less distracted than I was, since you didn't seem to notice what I wrote in boldface :)

Sorry, the title should have been: geometric intepretation of moments.
 

1. What is the geometric interpretation of a Taylor series?

The geometric interpretation of a Taylor series is that it represents a function as an infinite sum of polynomials. These polynomials approximate the original function at a specific point and can be used to estimate the behavior of the function at nearby points.

2. How is a Taylor series derived?

A Taylor series is derived by taking the derivatives of a function at a specific point and using those values to construct the coefficients of the polynomial terms in the series. The higher the degree of the polynomial, the more accurate the approximation will be.

3. What is the purpose of using a Taylor series?

The purpose of using a Taylor series is to approximate a function at a specific point and to estimate the behavior of the function at nearby points. It can also be used to simplify complex functions and make them more manageable for calculations.

4. Can a Taylor series be used to find the value of a function at a point outside of its domain?

No, a Taylor series can only approximate the behavior of a function within its domain. If a point is outside of the domain, the Taylor series will not accurately represent the function at that point.

5. How can the accuracy of a Taylor series be improved?

The accuracy of a Taylor series can be improved by including more terms in the series, as this will result in a higher degree polynomial and a closer approximation to the original function. Additionally, using a smaller interval around the point of approximation can also improve accuracy.

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