  We study antiferromagnetic spin chains with unfrustrated long-range
interactions that decays as a power law with exponent $\beta$, using the spin
wave approximation. We find for sufficiently large spin $S$, the Neel order is
stable at T=0 for $\beta < 3$, and survive up to a finite Neel temperature for
$\beta < 2$, validating the spin-wave approach in these regimes. We estimate
the critical values of $S$ and $T$ for the Neel order to be stable. The spin
wave spectra are found to be gapless but have non-linear momentum dependence at
long wave length, which is responsible for the suppression of quantum and
thermal fluctuations and stabilizing the Neel state. We also show that for
$\beta\le 1$ and for a large but finite-size system size $L$, the excitation
gap of the system approaches zero slower than $L^{-1}$, a behavior that is in
contrast to the Lieb-Schulz-Mattis theorem.
