MHB Does the Converse of the Ratio Test Always Hold for Convergent Series?

[alexmahone]

Is the converse of the ratio test true?

 

[Krizalid1]

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

 

[alexmahone]

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]

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]

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]

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 $\displaystyle\lim\left|\frac{a_{n+1}}{a_n}\right|\le 1$?