Which series can be truncated?

In summary, the Taylor series can be truncated due to the decreasing coefficients, while the Fourier series can also be truncated despite the coefficients not necessarily getting smaller. This is because in the Fourier series, the higher frequency components have very small amounts of energy and can be ignored for an approximation.
  • #1
HomogenousCow
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The taylor series can obviously be truncated, because the coeffecient of each series gets smaller and smaller due to the factorial.
However this is not the case with the fouriers series, there is no obvious reason why the coeffecients should get smaller and smaller.
So my question is, what kind of series can be truncated for an approximation?
 
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  • #2
Fourier series' absolutely can be truncated; how else would applied mathematics work with them?
 
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  • #3
I did not claim that it cannot be truncated, I am simply pointing out that even though it's terms do not get successively smaller, it can still be truncated.
 
  • #4
HomogenousCow said:
The taylor series can obviously be truncated, because the coeffecient of each series gets smaller and smaller due to the factorial.
That statement suggests you haven't studied the convergence of series. There are several things wrong with it - including the fact that a series doesn't necessarily converge even if the terms do "get smaller and smaller". But a thread on PF isn't the right place to explain something that takes a whole chapter in a math textbook - or even a whole textbook, depending how much detail you want to go into!

However this is not the case with the fouriers series, there is no obvious reason why the coeffecients should get smaller and smaller.
If the Fourier series represents something physical, the coefficients (squared) represent the amount of energy in each Fourier term. If the amount of energy in the system is finite, dividing the finite amount of energy into an infinite number of Fourier components means that almost all the components will have "very small" amounts of energy. Usually, the low frequency components are the only ones with "large" amounts of energy, so it's a good approximation to truncate the series and ignore all the high frequency components.
 
  • #5
Yes I understand that simply getting smaller does not garuntee convergence, the point I am making here is that because of the 1/n! behavior of the successive terms, when x is small this factor "drowns out" the monomial it is multiplied to, hence then it is obvious that we can truncate the series.
 

FAQ: Which series can be truncated?

What is truncation in a series?

Truncation in a series is the process of shortening the series by removing some of its terms. This is typically done to simplify the series or to make it more manageable for calculations.

What is the purpose of truncating a series?

The purpose of truncating a series is to reduce its complexity and make it easier to work with. Truncation is often used in mathematical equations and scientific calculations to simplify the series and make it more practical to use.

What are some common methods for truncating a series?

Some common methods for truncating a series include using a fixed number of terms, using a certain percentage of the series, or using a specific value as the cutoff point. The method used will depend on the specific series and its purpose.

What factors should be considered when deciding which series to truncate?

When deciding which series to truncate, factors such as the accuracy needed, the complexity of the series, and the desired level of precision should be considered. It is important to choose a truncation method that will not significantly affect the accuracy of the final result.

What are some possible consequences of truncating a series?

Truncating a series can lead to errors or inaccuracies in the final result, especially if the truncation method is not carefully chosen or if a large number of terms are removed. It is important to consider the potential consequences and choose the appropriate truncation method for the specific series and its intended use.

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