a systematic way a sequence of lower and upper bounds [@BO78]. This difficulty can be overcome by noticing that most of the quantities we are interested in, such as for example free energy or entropy or $q(1)$, do have a null derivative at $T=0$. Therefore an indication on the quality of the approximants can be obtained by analyzing the behaviour near $T=0$. For example, the free energy as a function of $T$ is reproduced quite well by many Padè approximants, even at very low orders, however some of these have a positive derivative at $T=0$ while others negative, see Fig. \[fig:freepad\]. By inspecting the figure we can safely assume that approximants with positive derivative give an upper bound, while those with negative derivative a lower bound, for the true free energy [@Note].

As a general fact we obtain that the best Padè approximants at a given order in $\tau$ are those with nearly the same degree of the numerator and the denominator. We stress, however, that as usual with asymptotic expansion an increase of the order in $\tau$ does not necessarily correspond to an improvement of the precision. With this procedure we obtain for the free energy an estimate with 