  We investigate the universality class of the finite-temperature phase
transition of the two-dimensional Ising model with the algebraically decaying
ferromagnetic long-range interaction, $J_{ij} = |\vec{r}_i
-\vec{r}_j|^{-(d+\sigma)}$, where $d$ (=2) is the dimension of the system and
$\sigma$ the decay exponent, by means of the order-$N$ cluster-algorithm Monte
Carlo method. In particular, we focus on the upper and lower critical decay
exponents, the boundaries between the mean-field-universality, intermediate,
and short-range-universality regimes. At the critical decay exponents, it is
found that the critical amplitude of the standard Binder ratio of magnetization
exhibits the extremely slow convergence as a function of the system size. We
propose more effective physical quantities, the combined Binder ratio and the
self-combined Binder ratio, both of which cancel the leading finite-size
corrections of the conventional Binder ratio. Utilizing these techniques, we
clearly demonstrate that in two dimensions the lower and upper critical decay
exponents are $\sigma = 1$ and 7/4, respectively, contrary to the recent Monte
Carlo and the renormalization-group studies [M. Picco, arXiv:1207.1018; T.
Blanchard, et al., Europhys. Lett. 101, 56003 (2013)].
