To show that a sequence of functions is a Cauchy sequence I am trying to prove:
Show that 
$$f_{n}(x)=\sum_{k=1}^{n} \frac{\sin(kx)}{k^{2/3}}$$ is Cauchy in $L^1[0,2\pi]$.
Any help will be appreciated. 
Thank you!
 A: Beware: overkill. The functions $g_k(x)=\frac{\sin(k x)}{k}$ are symmetric with respect to $x=\pi$, hence it is enough to prove that $\{f_n\}_{n\geq 1}$ is convergent in $ L^1\left(0,\frac{\pi}{2}\right)$. By the Fejer-Jackson inequality
$$ \forall x\in\left(0,\frac{\pi}{2}\right),\forall n\geq 1,\qquad G_n(x)\stackrel{\text{def}}{=}\sum_{k=1}^{n}g_k(x) \geq 0 \tag{1}$$
and by summation by parts
$$ \sum_{k=1}^{n} k^{1/3}g_k(x) = n^{1/3} G_n(x)-\sum_{k=1}^{n-1}G_k(x)\left((k+1)^{1/3}-k^{1/3}\right).\tag{2}$$
On the other hand
$$ \int_{0}^{\pi/2}G_m(x)\,dx = \sum_{k=1}^{m}\frac{2\sin^2\frac{\pi k}{4}}{k^2} \tag{3}$$
converges to a finite limit ($\frac{3\pi^2}{16}$) as $m\to +\infty$, as fast as $\int_{0}^{\pi/2}G_m(x)\,dx = \frac{3\pi^2}{16}-O\left(\frac{1}{m}\right)$,  and $(k+1)^{1/3}-k^{1/3}$ behaves like $\frac{1}{3k^{2/3}}$ for large values of $k$. Long story short: we may prove the convergence in $L^1$ like we prove the pointwise convergence, since the Fejer-Jackson inequality provides a non-negative bound that is simple and effective to exploit by summation by parts.
A: We first admit the following lemma:

For $\{c_k\}_{k\in \mathbb{Z}}\in \ell^2(\mathbb{Z})$, the function $f : [0, 2\pi]\to \mathbb{R}$ defined by $$f(x) := \sum_{k\in \mathbb{Z}} c_k\sin(kx)$$ is in $L^2([0, 2\pi])$.

Namely, the partial sums are Cauchy in $L^2([0, 2\pi])$, as $$\left\|\sum_{k=M}^N c_k\sin(kx)\right\|_{L^2([0, 2\pi])}^2 = \pi\sum_{k=M}^N c_k^2$$ due to the orthogonality of $\{\sin(kx)\}_{k=1}^{\infty}$ in $L^2([0, 2\pi])$ and the fact that $\int_0^{2\pi} \sin^2(kx)\,\mathrm{d}x = \pi$ for all $k\in \mathbb{Z}$. We let $c_k = \frac{1}{k^{2/3}}$ for $k\geq 1$ (and $c_k = 0$ otherwise) and note that $$\sum_{k\in \mathbb{Z}} c_k^2 = \sum_{k=1}^{\infty} \frac{1}{k^{4/3}} = \zeta(4/3)\approx 3.6$$ Therefore, $f(x) := \sum_{k=1}^{\infty} \frac{\sin(kx)}{k^{2/3}}$ is in $L^2([0, 2\pi])$. The Riesz-Fischer theorem then tells us that $f_n$ converges to $f$ in $L^2([0, 2\pi])$. As $[0, 2\pi]$ has finite measure, we have the inclusion $L^2([0, 2\pi])\subset L^1([0, 2\pi])$, so $f_n$ converges to $f$ in $L^1([0, 2\pi])$. This implies that $\{f_n\}_{n=1}^{\infty}$ is Cauchy in $L^1([0, 2\pi])$.
More generally, this works for any series $\sum_{k=1}^{\infty} \frac{\sin(kx)}{k^z}$ or $\sum_{k=1}^{\infty} \frac{\cos(kx)}{k^z}$ with $z > 1/2$ (i.e. such that $\zeta(2z)$ is finite).
