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We prove that in the limit where the beta function is dominated by the 1-loopcontribution (``large beta_0 limit'') diagonal Pad\'e Approximants (PA's) ofperturbative series become exactly renormalization scale (RS) independent. Thissymmetry suggest that diagonal PA's are resumming correctly contributions fromhigher order diagrams that are responsible for the renormalization of thecoupling-constant. Non-diagonal PA's are not exactly invariant, but generallyreduce the RS dependence as compared to partial-sums. In physical cases,higher-order corrections in the beta function break the symmetry softly,introducing a small scale and scheme dependence. We also compare the Pad\'eresummation with the BLM method. We find that in the large-N_f limit using theBLM scale is identical to resumming the series by a $x[0/n]$ non-diagonal PA.Comment: 25 pages, LateX. Replaced so that the figures would fit into the page  siz

Why Padé approximants reduce the renormalization-scale dependence in QFT