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

[zetafunction]

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

given a Fourier series of the form

\sum_{n=0}^{\infty}\frac{cos(nx)}{\sqrt{n}}

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

 

[mathman]

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

 

[AlephZero]

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.