The hydra-universe twisting its body scaled in stars.

The Proof of Stirling's approximation


When describing the time complexity of an algorithm, we often use Big O notation to describe the growth of computation with the size of the problem. For a factorial $n!$ of positive integers, this time complexity is $O(n)$ if a general recursive implementation is used, which represents a linear increase in computational effort as $n$ grows. When $n$ is large,...

The Proof of Wallis' formula


I read a question on Zhihu "Is there a problem with a result containing 'π' but completely unrelated to the circle?", and I thought of Wallis' formula, which was proposed by John Wallis (1616-1703). The main application still lies in the proof of Sterling's formula. Although Wallis' formula itself seems to be an ordinary infinite product, but his result expresses...