  We provide an exact infinite power series solution that describes the
trajectory of a nonlinear simple pendulum undergoing librating and rotating
motion 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 expansion
should be chosen to assure series convergence and maximize the domain of
convergence -- was not examined, and is provided here. By virtue of its
representation as an elliptic function, the pendulum function has singularities
that lie off of the real axis in the complex time plane. This, in turn, imposes
a radius of convergence on the physical problem in real time. By choosing the
expansion point at the top of the trajectory, the power series converges all
the way to the bottom of the trajectory without being affected by these
singularities. In constructing the series solution, we re-derive the
coefficients using an alternative approach that generalizes to other nonlinear
problems of mathematical physics. Additionally, we provide an exact resummation
of the pendulum series -- Motivated by the asymptotic approximant method given
in [Barlow et al., Q. J. Mech. Appl. Math., 70 (1) (2017), pp. 21-48] -- that
accelerates the series' convergence uniformly from the top to the bottom of the
trajectory. We also provide an accelerated exact resummation of the infinite
series representation for the elliptic integral used in calculating the period
of a pendulum's trajectory. This allows one to preserve analyticity in the use
of the period to extend the pendulum series for all time via symmetry.

Classical Analysis and ODEs