Strong asymptotic conditions (short guide to using summability methods)

The traditional wisdom that the Borel (non)summability (indicates) prevents the presence of non-perturbative effects and instability of a ground state is mostly incorrect. What is essential is the validity or violation of strong asymptotic conditions (SAC), which are formulated for a variety of regions. This is necessary, e.g., to prove (or disprove) the Callan-Symanzik assumption about the mass insertion in four-dimensional field theories. An example is shown of a formally Borel summable series violating SAC, and vice versa. (author) 1 fig., 31 refs