Not sure I get the Taylor Series

In summary, the Taylor series is a helpful tool for approximating a complex function in a local region, and it can be used to simplify the analysis of energy landscapes and other systems. While it does require knowledge of the function's derivatives, it is most useful when the variable is small and only a few terms are needed. Other series expansions, such as the Fourier series, may also be useful depending on the situation.
  • #1
fisico30
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not sure I get the Taylor Series...

Hello Everyone.

I understand that the taylor series approximate a function locally about a point, within the radius of convergence.
If we use the Taylor series it means that we do not know the function itself.

But to find the taylor series we need the derivatives of the function. and to have the derivatives we need the function itself...

where is the problem?

Also, we could represent a function is a certain interval of interest by using orthogonal functions (legendre, sines cosines, and any other orthogonal set). Why go with the Taylor series?
 
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  • #2


fisico30 said:
If we use the Taylor series it means that we do not know the function itself.

Not true! The Taylor series is useful in simplifying a known, complex function in a local region.

fisico30 said:
Why go with the Taylor series?

An example: the Taylor series tells us that any function is approximately quadratic around a minimum or maximum. This has tremendous implications when analyzing energy landscapes. Since a spring's energy is also quadratic around its equilibrium point, we can apply a lot of existing mathematics (e.g., simple harmonic motion) to systems near equilibrium.
 
  • #3


fisico30 said:
But to find the taylor series we need the derivatives of the function. and to have the derivatives we need the function itself...

It's most useful if "x" is "small", and you only need one or two terms to get a good approximation. I have heard (humourously) that all of theoretical physics consists of looking for "small parameters" so they can use Taylor series.

It's also useful if you want to represent a known/unknown function in another form which may cast the problem in a more familiar form.

fisico30 said:
Also, we could represent a function is a certain interval of interest by using orthogonal functions (legendre, sines cosines, and any other orthogonal set). Why go with the Taylor series?

All the various series expansions are potentially useful, and which one to use depends on the situation. For example, in describing sound hitting the ear, some variant of the Fourier series/transform is used most often because it's close to what the ear does.
 

What is the purpose of the Taylor Series?

The Taylor Series is a mathematical tool that allows us to approximate complex functions by using a series of simpler functions.

How is the Taylor Series calculated?

The Taylor Series is calculated by using the Taylor expansion formula, which involves taking the derivative of the function at a specific point and plugging it into the formula along with the value of the point.

What is the significance of the remainder term in the Taylor Series?

The remainder term in the Taylor Series represents the difference between the actual value of the function and the value approximated by the series. It allows us to determine the accuracy of the approximation.

Can the Taylor Series be used to approximate any function?

No, the Taylor Series can only be used to approximate functions that are infinitely differentiable, meaning they have derivatives of all orders at every point.

What are some real-world applications of the Taylor Series?

The Taylor Series is used in various fields such as physics, engineering, and economics to approximate complex functions and make predictions based on the approximated values.

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