Converse of the ratio test

Alexmahone

Active member
Is the converse of the ratio test true?

Krizalid

Active member
I don't think so. I think you can construct an easy counterexample. Care to imagine one?

Alexmahone

Active member
I don't think so. I think you can construct an easy counterexample. Care to imagine one?
0+0+0+... converges but the ratio is not defined.

I wonder if there are any non-trivial counterexamples.

HallsofIvy

Well-known member
MHB Math Helper
0+0+0+... converges but the ratio is not defined.

I wonder if there are any non-trivial counterexamples.
The "ratio test" says that if $lim \frac{a_{n+1}}{a_n}< 1$ then $\sum a_n$ converges.

The converse is "if $\sum a_n$ converged then $lim \frac{a_{n+1}}{a_n}< 1$".

Find a convergent series such that that limit is 1.

Also sprach Zarathustra

Member
0+0+0+... converges but the ratio is not defined.

I wonder if there are any non-trivial counterexamples.

Maybe...

$$a_n=\frac{1}{n(n-1)}$$

Alexmahone

Active member
Find a convergent series such that that limit is 1.
$\displaystyle\sum\frac{1}{n^2}$

So, is it safe to say that if a series converges, then $\displaystyle\lim\left|\frac{a_{n+1}}{a_n}\right|\le 1$?

• Also sprach Zarathustra