The bitterest tears shed over graves are for words left unsaid and deeds left undone.

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...