Using a data set of 6.32 fb$^{-1}$ of $e^+ e^-$ annihilation data collected with the BESIII detector at center-of-mass energies between 4178 and 4226 MeV, we have measured the absolute branching fraction of the leptonic decay $D_s^+ to tau^+ u_{tau}$ via $tau^+ to e^+ u_e bar{ u}_{tau}$ to be $mathcal{B}_{D_s^+ to tau^+ u_{tau}}=(5.27pm0.10pm0.12)times10^{-2}$, where the first uncertainty is statistical and the second is systematic. Combining with $f_{D_s^+}$ from Lattice quantum chromodynamics calculations or the $|V_{cs}|$ from the CKMfitter group, we extract $|V_{cs}|=0.978pm0.009pm0.012$ and $f_{D_s^+}= (251.1pm2.4pm3.0)$ MeV, respectively. These results are the most precise to date. Combining our result with the world averages of $mathcal{B}_{D_s^+ to tau^+ u_{tau}}$ and $mathcal{B}_{D_s^+ to mu^+ u_{mu}}$, we obtain the ratio of the branching fractions $mathcal{B}_{D_s^+ to tau^+ u_{tau}} / mathcal{B}_{D_s^+ to mu^+ u_{mu}} = 9.72pm0.37$, which is consistent with the standard model prediction of lepton flavor universality.
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