1. The problem statement, all variables and given/known data I know that the ratio test can be proved using the geometric series knowledge, but I'm looking for a way to prove the ratio test without the geometric series at all. The reason is that I'm proving the geometric series convergence with the ratio test, and my professor doesn't want me to use circular reasoning, because from what he remembered, he could only think of a proof with the geometric series. Do you guys know of any proof that doesn't involve the geometric series? I'm not trying to prove it myself, necessarily, just find the information somewhere. Any ideas?