$\displaystyle \sum \frac{e^{in\theta}}{\sqrt{n}}\\$
using Abel law
?
The 'Abel's test' for the convergence of a series is described here...
Abel's Uniform Convergence Test -- from Wolfram MathWorld
In the case You have proposed we can observe that...
a) the sequence $\displaystyle e^{in\theta}$ is not 'monotonic decreasing'...
b) the series $\displaystyle \sum_{n} \frac{1}{\sqrt{n}}$ diverges...
It seems that the result of the Abel's test in this case is negative...
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$
The series you have proposed is the 'powers series'...
$\displaystyle \sum_{n=1}^{\infty} \frac{z^{n}}{\sqrt{n}}$ (1)
... computed in $\displaystyle z=e^{i\theta}$ , i.e. on the circle with center in $\displaystyle z=0$ and radious $\displaystyle 1$. It is easy to demonstrate that (1) converges for $\displaystyle |z|<1$, i.e. for all points internal to that circle. A general criterion to extablish what happens on the circle of convergence doesn't exist. In this case we can say that the (1) diverges for $\displaystyle \theta=0$... for other points a detailed study is required...
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$