Resummation Borel

The convergence of perturbation theory is discussed in [ZJ89], where the method of Borel transformation and resummation is also explained. 

The Pade-Borel procedure can be optimized by introducing an additional fit parameter p to the Borel transformation. Substituting the factorial i by the Euler gamma-function r(i -l-p -I-1) and inserting an additional factor f into the integral (90), one defines the Pad -Borel-Leroy resummation procedure. 

In order to suit the resummation procedure (88)-(90) for functions that depend on several variables one should change the first step (88) for example, for the two-variable case one defines the Borel image by [92] ... 

It is obvious that an initial sum can be resummed in different ways. Apart from the Leroy fit parameter p mentioned above some arbitrariness arises from the different types of rational approximants one may construct. For instance, within the two-loop approximation the method of Pade-Borel resummation of a resolvent series can be done using either the [0/2] or the [1/1] approximants. The Chisholm-Borel approximation implies even more arbitrariness and demands a careful analysis of the approximants to be chosen. 

This assumption implies that the sum in this form converges to the trae (but unknown) value of the groimd-state eneigy. In most cases, this assumption is actually not satisfied. Starting at a certain order the correction can become laiger than the first-order contribution - the series diverges. Tmncating such a series after it has swapped phases can only yield reasonable results, if resummation methods (Fade, Borel, variational approaches, etc.) are applied which are based on the fact that the trae result must be finite. 

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