Root test inconclusive implies ratio test inconclusive

Question

I’m a high school student and teaching myself a tiny bit of maths. I stumbled over a problem, I can’t find a proof for (and neither found a proof for it in the internet or in books, but I guess there is one).

If the root test is inconclusive, then the ratio test is inconclusive, too ( in other words: if the root test equals one, then the ratio test equals one).

Is this statement true? And is there a proof for it?

Any help is appreciated.

(The point of the proof should be showing that the root test is stronger than the ratio test. I know examples for this, but can't find a proof (maybe proof by contradiction?).)

What I want to show is:

$ \limsup \sqrt[n]{|a_n|}=1 \implies \lim |\frac{a_{n+1}}{a_n}|=1$

Does this answer your question? Inequality involving $\limsup$ and $\liminf$: $ \liminf(a_{n+1}/a_n) \le \liminf((a_n)^{(1/n)}) \le \limsup((a_n)^{(1/n)}) \le \limsup(a_{n+1}/a_n)$ — Aaratrick, Jun 23 '22 at 07:50
Only partly (or maybe I just don’t see it). The conclusion I can make from this, is just that if I the root test equals 1, then the ratio test could equal one (or be greater). — Marko, Jun 23 '22 at 10:24
See also https://math.stackexchange.com/questions/758957/relations-between-the-root-test-and-the-ratio-test and also https://math.stackexchange.com/questions/2434390/ratio-test-versus-root-test and also https://math.stackexchange.com/questions/4166260/ratio-test-and-root-test — Gerry Myerson, Jun 23 '22 at 13:40
But how does this prove my statement $ lim sup \sqrt[n]{|a_n|}=1 \implies lim |\frac{a_{n+1}}{a_n}|=1$ ? — Marko, Jun 23 '22 at 14:16

Root test inconclusive implies ratio test inconclusive

0 Answers0