  The production of $B^{\pm}$ mesons is studied in $pp$ collisions at
centre-of-mass energies of 7 and 13 TeV, using $B^{\pm}\rightarrow J/\psi
K^{\pm}$ decays and data samples corresponding to 1.0 fb$^{-1}$ and 0.3
fb$^{-1}$, respectively. The production cross-sections summed over both charges
and integrated over the transverse momentum range $0<p_{\text{T}}< 40$ GeV/$c$
and the rapidity range $2.0<y<4.5$ are measured to be $\sigma(pp \rightarrow
B^{\pm} X, \sqrt{s} = \text{7 TeV}) = 43.0 \pm 0.2 \pm 2.5 \pm 1.7\mu b,$
$\sigma(pp \rightarrow B^{\pm} X, \sqrt{s} = \text{13 TeV}) = 86.6 \pm 0.5 \pm
5.4 \pm 3.4\mu b,$ where the first uncertainties are statistical, the second
are systematic, and the third are due to the limited knowledge of the
$B^{\pm}\rightarrow J/\psi K^{\pm}$ branching fraction. The ratio of the
cross-section at 13 TeV to that at 7 TeV is determined to be
$2.02\pm0.02\text{(stat)}\pm0.12\text{(syst)}$. Differential cross-sections are
also reported as functions of $p_{\text{T}}$ and $y$. All results are in
agreement with theoretical calculations based on the state-of-art fixed
next-to-leading order quantum chromodynamics.
