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We provide an exact infinite power series solution that describes thetrajectory of a nonlinear simple pendulum undergoing librating and rotatingmotion for all time. Although the series coefficients were previously given in[V. Fair\'en, V. L\'opez, and L. Conde, Am. J. Phys 56 (1), (1988), pp. 57-61],the series itself -- as well as the optimal location about which an expansionshould be chosen to assure series convergence and maximize the domain ofconvergence -- was not examined, and is provided here. By virtue of itsrepresentation as an elliptic function, the pendulum function has singularitiesthat lie off of the real axis in the complex time plane. This, in turn, imposesa radius of convergence on the physical problem in real time. By choosing theexpansion point at the top of the trajectory, the power series converges allthe way to the bottom of the trajectory without being affected by thesesingularities. In constructing the series solution, we re-derive thecoefficients using an alternative approach that generalizes to other nonlinearproblems of mathematical physics. Additionally, we provide an exact resummationof the pendulum series -- Motivated by the asymptotic approximant method givenin [Barlow et al., Q. J. Mech. Appl. Math., 70 (1) (2017), pp. 21-48] -- thataccelerates the series' convergence uniformly from the top to the bottom of thetrajectory. We also provide an accelerated exact resummation of the infiniteseries representation for the elliptic integral used in calculating the periodof a pendulum's trajectory. This allows one to preserve analyticity in the useof the period to extend the pendulum series for all time via symmetry.

On the trajectory of the nonlinear pendulum: Exact analytic solutions via power series