The harmonic oscillator approximation gives $\epsilon_{vib}=\left(v+\frac{1}{2}\right)h\nu$ where $v$ is the vibrational quantum number ranging from $[0,\infty ]$ and there is no degeneracy.
It is usual in statistical mechanics to take the lowest energy level to be zero $\epsilon_{vib}=\left(v+\frac{1}{2}\right)h\nu-\frac{1}{2}h\ nu=vh\nu$
\begin{equation} q_{vib}=\sum_{v}e^{-\beta\epsilon_{vib,v}}=\sum_{v=0}^{\infty}e^{-\beta vh\ nu}=\sum_{0}^{\infty}e^{-\frac{h\nu}{kT}v}=\sum_{0}^{\infty}e^{-\frac{v\theta_ {vib}}{T}} \end{equation} Where $\theta_{vib}=\frac{h\nu}{k}$ called characteristic vibrational temperature.
Using the series expansion $\sum_{n=0}^{\infty}x^n=1+x+x^2+x^3+......=\frac{1}{1-x }$, the vibrational partitioning function can be written as follows: \begin{equation} q_{vib}=\frac{1}{1-e^{-\theta_{vib}/T}} \end{equation} This equation is valid as long as the temperature is not very high.