is in very good agreement to the experimental value, $(1.39\pm 0.09)\times 10^2$, recently reported by Avila [*et al.*]{} [@aetal-prl15].

[c|cccc]{} & $|-\frac{1}{a_l}|$ & $|\frac12(r_l-\tilde{r}_l)k_G^2|$ & $|-\frac14(P_l-\tilde{P}_l)k_G^4|$ & $|(Q_l-\tilde{Q}_l)k_G^6|$ $S2$ & 1 & 0.276 & 0.012 & 0.004 $P2$ & 0.154 & 1 & 0.215 & 0.016 $D2$ & 1 & 0.946 & 0.316 & 0.031 $F2$ & 1 & 0.195 & 0.023 & 0.002

To examine the convergence of the power series in terms of $k^2$ at $k=k_G$, we add the effective range terms and those from the $2\kappa ReH_l(k)$ functions together. In Table \[table;ratios\], we show the ratios of the terms after normalizing those terms by $-1/a_l$ for $l=0,2,3$ and by $\frac12(r_1-\tilde{r}_1)k_G^2$ for $l=1$ because of their dominance where the effective range parameters fitted by using the data sets, $S2$, $P2$, $D2$, $F2$, are used. As discussed above, the expansion parameter at $T_G$ is $Q\sim 1/3$, so the $k_G^2$, $k_G^4$, $k_G^6$ terms are expected to be a few tenth, a few hundredth, a few thousandth to the leading order terms, respectively. We find good convergence of the power series for $l=0,3$ at $k=k_G$. On the other hand, the $-1/a_1$ term is small compared to the $\frac12(r_1-\tilde{r}_1)k_G^2$ term for $l=1$ 