I have encountered a limiting process on some infinite series. So, I would like to ask:

QUESTION. Assume $n$ is an even positive integer. Is this true? $$\lim_{r\rightarrow1^{-}}\sum_{j=1}^{\infty}\left(\sqrt{\frac2n}-\cos\left(\frac{\pi j^2}n\right) -\sin\left(\frac{\pi j^2}n\right)\right)r^j=\frac12\left(1-\sqrt{\frac2n}\right).$$