Stirling’s Approximation Formula and Improvements

Stirling’s approximation formula for n!, the gamma function as an extension of the factorial to the complex numbers, improvements of Stirling’s approximation up to the twenty-first century, the Gaussian integral, limits, sequences, series, infinite products, infinite integrals, Taylor series, the standard normal distribution, the gamma distribution, the exponential distribution, the chi-square distribution, evaluating zeta(2), male births, carefree couples, and the probability that two randomly chosen natural numbers are coprime.