In mathematics, Stirling's approximation (or Stirling's formula) is an approximation for factorials. It is a very powerful approximation, leading to accurate results even for small values of n. It is named after James Stirling, though it was first stated by Abraham de Moivre. The formula as typically used in applications is (in big O notation). The next term in the O(ln n) is 1/2ln(2πn); a more precise variant of the formula is therefore It is also possible to give a version of Stirling's formula with bounds valid for all positive integers n, rather than asymptotics: one has for all positive...

Penguins (order Sphenisciformes, family Spheniscidae) are a group of aquatic, flightless birds living almost exclusively in the Southern Hemisphere, especially in Antarctica. Highly adapted for life in the water, penguins have countershaded dark and white plumage, and their wings have evolved into flippers. Most penguins feed on krill, fish, squid and other forms of sealife caught while swimming underwater. They spend about half of their lives on land and half in the oceans. Although all penguin species are native to the Southern Hemisphere, they are not found only in cold climates, such as...