Proof of Ratio Test on Infinite Series

[Fooze]

Homework Statement

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?

 

[snipez90]

I think you'll be hard pressed to find such a proof. The motivation behind the proof of the ratio test is to use the comparison test, which requires that we actually know a convergent series to begin with. Since it's easy to show the convergence of the geometric series using basic algebra and limits, there is no need to believe that any circular reasoning will result. I think you can prove the ratio test from the root test, but iirc, an easy proof of the root test also depends on comparison with the geometric series.

 

[Fooze]

Yes, that is what I was figuring... I was just hoping to prove my professor wrong. ;)

If anyone has any other thoughts though, please do let me know!

 

[n!kofeyn]

snipez90 is exactly right. Here are some references if you wanted to see the proofs in action.

p. 190-193 of Mathematical Analysis by Tom Apostol
p. 60-67 of Principles of Mathematical Analysis by Walter Rudin
p. 161-162 of Advanced Calculus by Creighton Buck