  We prove that in the limit where the beta function is dominated by the 1-loop
contribution (``large beta_0 limit'') diagonal Pad\'e Approximants (PA's) of
perturbative series become exactly renormalization scale (RS) independent. This
symmetry suggest that diagonal PA's are resumming correctly contributions from
higher order diagrams that are responsible for the renormalization of the
coupling-constant. Non-diagonal PA's are not exactly invariant, but generally
reduce 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\'e
resummation with the BLM method. We find that in the large-N_f limit using the
BLM scale is identical to resumming the series by a $x[0/n]$ non-diagonal PA.
