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\[ \rho_n(t) = \frac{-i}{\hbar}\int_{0}^{t}{d\tau[H(\tau),\rho_{n-1}(\tau)]} \]  By differentiating both sides, I define  an equation of motion for the n\ts{th} order perturbation is recursively defined: perturbation:  \[ \pd{\rho_n(t)}{t} = \frac{-i}{\hbar}[H(t),\rho_{n-1}(t)] \]