Find the $2\pi$-periodic $C^1$ function from the given Fourier coefficients Which $2\pi$-periodic $C^1$ function has the Fourier coefficients
$$
c_n = \begin{cases}
  ne^{-n}, & n\geq 0\\
  0, & n < 0 
\end{cases}
$$
The function that we want to find can be described as
$$
u(x) = \sum_{-\infty}^{\infty}c_ne^{inx} = \sum_{0}^{\infty}ne^{-n}e^{inx} + \sum_{-\infty}^{0}0 = \sum_{0}^{\infty}ne^{-n}e^{inx}
$$
but not sure where to go from here. Help appreciated.

So you have 
$$\sum_{n=0}^\infty n \left(e^{ix-1}\right)^n$$
Note that this series converges for every $x$ since $\left|e^{ix-1}\right| = \frac 1e < 1$.
To find to which values it converges to, you start from the series (valid for $|t | < 1$)
$$\sum_{n=0}^\infty t^n = \frac 1{1-t}$$
Differentiating wrt to $t$ both sides and multiplying by $t$, you get 
$$\sum_{n=0}^\infty nt^n = \frac t{(1-t)^2}$$
Now plug in $t = e^{ix-1}$ and you are done.
