When are the Fourier coefficients of $2\pi$-periodic functions summable? More specifically, which ones of $f(x) = (\cos x)^{100}$, $g(x) = \sin(\tan x)$, and $$ h(x) = \begin{cases} x+\pi, & : x\in[-\pi,0] \\ \pi-x, & : x\in (0,\pi] \end{cases} $$
are absolutely summable?
As for summability, use Dini's theorem, or at least the one I learned in my real analysis course (I don't like Wikipedia's convoluted explanation of it). If the function is $L_1$, roughly meaning that it has a finite integral on the interval, and is Lipschitz continuous on the interval, then the partial sums converge to the function value, which hopefully is finite.
So then $f$ and $h$ have convergence everywhere (due to Lipschitz), but $g$ has continuity problems at $\pm\pi/2$, so Dini can't be applied there. At a glance, I'm quite sure convergence fails for $g$ at those spots.