Thermal partition function of a quantum harmonic oscillator ($Z(\beta ,\omega )=[2\sinh (\beta \omega /2){]}^{-1}$)

For a quantum harmonic oscillator with $\hslash =1$, $\omega >0$ and inverse temperature $\beta >0$, the canonical partition function is ${\sum }_{r\ge 0}{e}^{-\beta \omega (r+1/2)}$. The geometric series gives the displayed formula. Keeping the zero-point energy is essential to its prefactor.