I am trying to understand what my textbook is saying about the minimum number of coefficients we should calculate to see convergence. Here is what it says (specific to an even function) :
Given that $\lambda_n = 2L/n$ is the wavelength of the nth term, we see that the wavelength decreases like $1/n$
Ok, got it.
Knowing this, let's think for a minute how this integral will evaluate with different $f(x)$ and different $n$'s. If $f(x)$ is nearly constant, then $a_n$ will evaluate to a very small number for $n > 0$ because in the limit $f(x) = c$ and $c\int_0^L cos(n\pi x/L) = 0$
Ok, I mostly get it. $f(x)$ is "constant" if we zoom in enough, so this integral gets small.
Now let's imagine what would happen if $f(x)$ is smooth and oscillates gently two times on the integral, and we choose a very large value of n, say $n=100$ with correspondingly small wavelength. Then if we split the integral to be the sum of integrals over each wavelength, each integral will evaluate to a number very close to zero again because over each oscillation period, the function is nearly constant. This is the essence of convergence of a Fourier series; oscillatory terms that vary on the same scales as $f(x)$ contribute most to the Fourier sum.
So, what is he talking about in this bit? So for the coefficient $a_{100}$, what are we dealing with here when he says to "split the integral to be a sum of integrals over each wavelength"? This particular coefficient only has a single wavelength. Is he meaning that we subdivide the integral by that wavelength, meaning that each one is super, super zoomed in?
Also, what is the point of introducing two oscillations in the interval? What exactly does that have to do with this example? Is he just trying to emphasize that there are no small features to resolve?
Thanks.