Which series can be truncated?

[HomogenousCow]

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?

 

[Number Nine]

Fourier series' absolutely can be truncated; how else would applied mathematics work with them?

 

[HomogenousCow]

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.

 

[AlephZero]

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.

 

[HomogenousCow]

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.