Can we simply truncate a Fourier series if it is divergent?

In summary, truncating a Fourier series can result in finite results, but it does not guarantee convergence to a specific value. It is important to consider the intended representation and purpose of the series before truncating it.
  • #1
zetafunction
391
0
can we simply truncate a Fourier series if it is divergent??

given a Fourier series of the form

[tex] \sum_{n=0}^{\infty}\frac{cos(nx)}{\sqrt{n}}[/tex]

can i simply truncate this series up to some number finite N so i can get finite results ?? thanks.
 
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  • #2


You can always truncate the series to get a finite result. However the result as a function of N does not converge to anything.
 
  • #3


Maybe a more fundamental question is "what is your series supposed to represent?" For eaxmple, the energy (measured as the function squared) is ##\sum (1/n)## which is infinite.

As mathman said, you can do anything you like mathematically with a finite number of terms, but whether the result means anything is another question.
 

FAQ: Can we simply truncate a Fourier series if it is divergent?

Can a Fourier series be truncated if it is divergent?

Yes, a Fourier series can be truncated if it is divergent. Truncating a series means only considering a finite number of terms in the series, which can help simplify calculations and provide a good approximation of the original function.

How do we know when to truncate a Fourier series?

The decision to truncate a Fourier series depends on the desired level of accuracy. Generally, truncating a series after a certain number of terms can provide a good approximation of the original function, but more terms may be needed for higher accuracy.

What happens if we truncate a Fourier series too early?

If a Fourier series is truncated too early, the resulting approximation may not accurately represent the original function. This can lead to errors or inaccuracies in calculations or analysis.

Can we always truncate a Fourier series if it is divergent?

No, not all Fourier series can be truncated if they are divergent. Some functions may have a non-trivial Fourier series that cannot be accurately represented by truncation. In these cases, other methods may be needed to analyze the function.

How does truncating a Fourier series affect its convergence?

Truncating a Fourier series does not affect its convergence. The convergence of a series is determined by the original function and the properties of the Fourier coefficients, not by the number of terms considered in the series. However, truncating a series may affect the accuracy of the resulting approximation.

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