Question

Why is the Ratio Test for convergence is true?

Answer 1

It is a special case of the comparison test, comparing with the geometric series.

Answer 2

I watched http://www.youtube.com/watch?v=rUis1mSzwyA and I could follow the reasoning until the part (04:55) where she explains \[\frac{a_{n+1}}{r_{n+1}}<\frac{a_{n}}{r_{n}}\] implies \[a_n < \mathrm{k} r^n\] can anyone explain how she got that expression to me? thanks!

Answer 3

whoops I wrote it wrong, it should be \[\frac{a_{n+1}}{r^{n+1}}< \frac{a_n}{r^n}\]