Consider the series
$$
\sum_{n=1}^{\infty} ( \\{  n \xi \\} - \frac{1}{2})
$$
where $\\{ \\}$ denotes the fractional part and $\xi$ is some irrational number. It clearly does not converge but can we show that it does not tend to $\infty$ or $-\infty$? Would it also be true that it bounded?

The problem seems to be related to equidistributed sequences. Since ${(\\{ n \xi  \\} )}_{n \geq 1}$ is equidistributed we have that 

$$
\frac{1}{N}\sum_{n=1}^{N}(\\{n\xi\\}- \frac{1}{2}) \to 0
$$
as $N \to \infty$. 
However, just the equidistribution property cannot be enough. Consider for example a sequence $(a_n)$ given by $a_n = 1$ if $n$ is odd or a power of $2$ and $a_n= -1$ otherwise. Then $(a_n)$ is equidistributed on $ \\{ -1,1 \\}$ but 

$$
\sum_{n=0}^{\infty} a_n = \infty.
$$