We first take up the question of how good periodic Fourier series is. How many terms are needed to get a good approximation to a function? This is equivalent to asking how fast the coefficients $\hat f_n$ decay with $n$. The smoother the function, the faster the decay:

1. If $f(x)$ has a jump discontinuity in the $k$-th derivative and nothing worse, then $\hat f_n$ goes as $n^{-(k+1)}$.
2. If $f(x)$ is infinitely differentiable, then $\hat f_n$ decays faster than any power of $n$. No matter what exponent $p$ we pick, $\lim_{n \to \pm \infty} n^p \hat f_n = 0$.
3. If $f(x)$ is analytic, meaning that it can be written as a convergent power series, then $\hat f_n$ decays exponentially.

Now that we have the convergence rates for periodic Fourier series, we can go back and study convergence for sine Fourier series. If $f(x)$ is a function on $[0,L]$ with $f(0)=f(L)=0$ and $$ f(x) = \sum_{n=1}^\infty c_n \sin(n\pi x/L),$$ then we construct a new function $\tilde f(x)$ on the real line with $$\tilde f(x) = \begin{cases} f(x) & 0 \le x \le L \cr -f(-x) & -L \le x \le 0 \cr f(x+2L) & \hbox{always} \end{cases} $$ Since $\tilde f(x)$ is periodic, we can expand it using periodic Fourier series: $$\tilde f(x) = \frac{a_0}{2} + \sum_{n=1}^\infty a_n \cos(2\pi n x/2L) + b_n \sin(2\pi n x/2L),$$ where we have $2L$ in the denominator because $\tilde f(x)$ is periodic with period $2L$. The coefficients $a_n$ are all zero by symmetry, since $\tilde f(x)$ is an odd function, and the coefficients $b_n$ are exactly the same as the coefficients $c_n$ in the sine expansion of $f(x)$. Since we know how fast the $b_n$'s decay, we know how fast the $c_n$'s decay. This is explained in the second video: