Absolute convergence of fourier series Let's say I have a piecewise continuous function which has the fourier series  $\sum_n\ c_{n}e^{inx}$ and I assume that $ \sum_n\ n|c_{n}|$ converges, then I know the following holds:
The fourier series of the derivatives $\sum_n\ inc_{n}e^{inx}$ is absolutely convergent and thus from the Weierstrass M-test I know $\sum_n\ inc_{n}e^{inx}$ is uniformly convergent. 
Thus, $ f' (x)=\sum_n\ inc_{n}e^{inx}$ is a continuous $2 \pi $-periodic function.
We can say then that $ f(x)=\sum_n\ c_{n}e^{inx}$ is a continuously differentiable $2 \pi $-periodic function.
My question is this:
Does the converse also holds?
To be more precise: 
If  $  f(x)=\sum_n\ c_{n}e^{inx}$ is a continuously differentiable $2 \pi $-periodic function, does it follow that $\sum_n\ n|c_{n}|$  converges?
My thought on the matter are such:
I think the converse does not hold, since we've used the Weierstrass M-test which is kind of a one-way ticket.
I attempted at constructing a counter-example.
Since I want $\sum_n\ n|c_{n}|$ to diverge, it suffices to find $c_{n}$ such that $c_{n}   \sim  \frac{1}{ n^{2} } $.
Thus, if I look at  $ f' (x)=\sum_n\ inc_{n}e^{inx}$,
it is clear that I need a continuous $2 \pi $-periodic function such that its fourier coefficients decay rate is $\frac{1}{ n } $.
Unfortunately, every continuous function I tried had a fourier coefficients decay rate of $\frac{1}{  {n^{2} }  } $.
The ones that I know of that have a decay rate of $   \frac{1}{ n }   $ are discontinuous functions.
Your thoughts on the matter are greatly appreciated!

Suppose $h$ is a periodic continuous function $[0,2\pi]$ for which $\sum_n |c_n| =\infty$. Define
$$
      f(x) = \int_0^x \left[h(t)-\frac{1}{2\pi}\int_0^{2\pi}h(u)du\right]dt.
$$
Then $f$ is continuously differentiable and periodic on $[0,2\pi]$ with Fourier coefficients $c_n(f)$ for which
\begin{align}
  inc_n(f)
     & = -\frac{1}{2\pi}\int_{0}^{2\pi}f(y)(-ine^{-iny})dy \\
     & = -\frac{1}{2\pi}\int_{0}^{2\pi}f(y)\frac{d}{dy}e^{-iny}dy \\
     & = \frac{1}{2\pi}\int_{0}^{2\pi}h(y)e^{-iny}dy = c_n(h),\;\;\; n\ne 0.
\end{align}
So your problem reduces to considering the Fourier series of a periodic continuous function $h$. It is known that there is a periodic continuous function $h$ such that the Fourier series for $h$ diverges at $0$. Such an $h$ provides the counter example that you want because $\sum_n|c_n(h)| =\infty$ must occur.
