Fractional Brownian motions (fBM) and related processes are widely used in financial modeling to capture the complicated dependence structure of the volatility. In this paper, we analyze an infinite series representation of fBM proposed in (Dzhaparidze and Van Zanten 2004) and establish an almost sure convergence rate of the series representation. The rate is also shown to be optimal. We then demonstrate how the strong convergence rate result can be applied to construct simulation algorithms with path-by-path error guarantees.