Theorem: The series $\psi_s(z)$ converges absolutely in the following cases:

1. when $z=\dfrac{p}{q}$ with $q$ odd, $s>1$
2. when $z$ algebraic irrational number and $s >2$
3. when $z$ is algebraic irrational and $s=2$.

Theorem: Let $z$ be irrational, $k \geq \dfrac{1}{2}$, and $\dfrac{p}{q}$ be a rational approximation to $z$ in reduced form for which $$\left| z - \dfrac{p}{q} \right|< \dfrac{k}{q^2}.$$ Then either $\dfrac{p}{q}$ is a convergent $\dfrac{p_{\ell}}{q_{\ell}}$ to $z$, or $$\dfrac{p}{q} = \dfrac{ap_{\ell}+bp_{\ell-1}}{aq_{\ell}+bq_{\ell-1}}; |a|,|b|<2k,$$ where $a$ and $b$ are integers.