We present a simple derivation of the mean square end-to-end distance 〈R2〉 of a linear flexible chain as a perturbation series in the dimensionless excluded volume parameter zd. Our results, to orders six and four in space dimension d=3 and 2, respectively, are 〈R2〉= Ll[1+ (4)/(3) z3−2.075 385 396 z23+6.296 879 676 z33 GFIX−25.057 250 72 z43+116.134 785 z53 GFIX−594.716 63 z63+⋅⋅⋅], d=3, 〈R2〉= Ll[1+ 1/2 z2−0.121 545 25 z22+0.026 631 36 z32 GFIX−0.132 236 03 z42+⋅⋅⋅], d=2, where z3=(3/2πl)3/2 wL1/2 and z2=wL/πl with L the contour length of the chain, l the effective or Kuhn segment length, and wl2 the effective binary cluster integral for a pair of segments. Our method uses in an essential way Laplace transforms with respect to the contour length L; the resulting graphical expansion, when combined with the field theoretical methods, is far simpler than that in the conventional cluster expansion approach. Furthermore, we prove that 〈R2〉 is free of ln L terms to all orders in the perturbation theory in both d=2 and 3.