What can we say about $\frac{1}{2(1-z)}-\sum_{n=0}^\infty\{n\sqrt{2}\}z^n$ in the vicinity of $z=1$?

Question

Let $\{x\}$ denote the fractional part of $x$, what can we say about $$\frac{1}{2(1-z)}-\sum_{n=0}^\infty\{n\sqrt{2}\}z^n$$ in the vicinity of $z=1$? It seems that there is no limit, but the partial sums seem bounded.