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We revisit the expansion recently proposed by Pulvirenti and Tsagkarogiannisfor a system of $N$ continuous particles in the canonical ensemble. Under thesole assumption that the particles interact via a tempered and stable pairpotential and are subjected to the usual free boundary conditions, we show theanalyticity of the Helmholtz free energy at low densities and, using thePenrose tree graph identity, we establish a lower bound for the convergenceradius which happens to be identical to the lower bound of the convergenceradius of the virial series in the grand canonical ensemble established byLebowitz and Penrose in 1964. We also show that the (Helmholtz) free energy canbe written as a series in power of the density whose $k$ order coefficientcoincides, modulo terms $o(N)/N$, with the $k$-order virial coefficient dividedby $k+1$, according to its expression in terms of the $m$-order (with $m\lek+1$) simply connected cluster integrals first given by Mayer in 1942. Wefinally give an upper bound for the $k$-order virial coefficient which slightlyimproves, at high temperatures, the bound obtained by Lebowitz and Penrose.

Continuos particles in the Canonical Ensemble as an abstract polymer gas