Intro to the Ionization Energy of Atomic Hydrogen

Introduction
In previous articles relating to various transition energies in Hydrogen, Helium and Deuterium we have employed the following formula for electron energy given a particular primary quantum number n:
$$ E_{n}=\mu c^2\sqrt{1-\frac{Z^2\alpha^2}{n^2}} $$
where ## \alpha ## is the fine structure constant and ## \mu ## the reduced electron mass for a single electron bound to whichever nucleus. Z=1 for Hydrogen and Deuterium, Z=2 for Helium. Reduced mass is calculated from electron mass and nuclear mass as follows: $$\mu = \frac{m_e\times m_{n}}{m_e+m_{n}} $$.
Calculation of Ionization Energy: Atomic Hydrogen
Perhaps one of the simplest applications of this formula is the determination of the ionization energy of...

as many significant figures as Wolfram Alpha is prepared to calculate!