Through the half-unit circle area computation using the integration of the corresponding curve power series representation, we deduce a slow converging positive infinite series to $\pi$. However, by studying the remainder of that series we establish sufficiently close inequalities with equivalent lower and upper bound terms allowing us to estimate, more precisely, how the series approaches $\pi$. We use the obtained inequalities to compute up to four-digit denominator, what are in this sense, the best rational numbers that can replace $\pi$. It turns out that the well-known convergents of the continued fraction of $\pi$, $22/7$ and $355/113$ called, respectively, Yuel\"{u} and Mil\"{u} in China are the only ones found. Thus we apply a similar process to find rational estimations to $\pi+e$ where $e$ is taken as the power series of the exponential function evaluated at $1$. For rational numbers with denominators less than $2000$, the convergent $920/157$ of the continued fraction of $\pi+e$ turns out to be the only rational number of this type.
Stable and Holomorphic Implementation of Complex Functions Using a Unit Circle-Based Transform
