  We investigate many-body localization in the chain of interacting spins with
a transverse power-law interaction, $J_{0}/r^{\alpha}$, and random on-site
potentials, $\phi_i \in \left(-W/2,W/2\right)$, in the long-range limit,
$\alpha< 3/2$, which has been recently examined experimentally on trapped ions.
The many-body localization threshold is characterized by the critical
disordering, $W_c$, which separates localized ($W > W_c$) and chaotic ($W <
W_c$) phases. Using the analysis of the instability of localized states with
respect to resonant interactions complemented by numerical finite size scaling,
we show that the critical disordering scales with the number of spins, $N$, as
$W_c \approx [1.37 J_{0}/(4/3 - \alpha)]N^{4/3 - \alpha} \ln N$ for $0 < \alpha
\leq 1$, and as $W_c \approx [J_{0}/(1-2\alpha/3)]N^{1-2\alpha/3} \ln^{2/3} N$
for $1 < \alpha < 3/2$ while the transition width scales as $\sigma_{W} \propto
W_{c}/N$. We use this result to predict the spin long-term evolution for a very
large number of spins ($N = 50$), inaccessible for exact diagonalization, and
to suggest the rescaling of hopping interaction with the system size to attain
the localization transition at finite disordering in the thermodynamic limit of
infinite number of spins.
