Concerning the Wikipedia article "root test", this convergence criterion for series was first proved by Augustin-Louis Cauchy. What is the name of the treatise where he proved the root test? Which page?
It's Theorem 1 from Chapter 6 Section 2 of Cauchy's Cours d'analyse, published in 1821. You can read the original at archive.org or have a look at the 2009 English translation by Rob Bradley and Ed Sandifer (published by Springer).