How Isaac Newton Discovered the Binomial Power Series

Steven Strogatz in Quanta:

Isaac Newton was not known for his generosity of spirit, and his disdain for his rivals was legendary. But in one letter to his competitor Gottfried Leibniz, now known as the Epistola Posterior, Newton comes off as nostalgic and almost friendly. In it, he tells a story from his student days, when he was just beginning to learn mathematics. He recounts how he made a major discovery equating areas under curves with infinite sums by a process of guessing and checking. His reasoning in the letter is so charming and accessible, it reminds me of the pattern-guessing games little kids like to play.

It all began when young Newton read John Wallis’ Arithmetica Infinitorum, a seminal work of 17th-century math. Wallis included a novel and inductive method of determining the value of pi, and Newton wanted to devise something similar.

More here.