Past exam of the quantum Condensed Matter Field Theory course of the University of Cambridge / 2014 / Paper 1 / 2 / b / Solution

The endpoint integral is a Gaussian with coefficient $A=(m\omega /\hslash )\tanh (x/2)$. ThereforeIt agrees with ${\sum }_{n\ge 0}{e}^{-x(n+1/2)}$, tends to $1/x$ at high temperature, and to $e^{-x/2}$ at low temperature. The last limit retains the oscillator zero-point energy.