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#### Alexmahone

##### Active member

- Jan 26, 2012

- 268

Is the converse of the ratio test true?

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- Jan 26, 2012

- 268

Is the converse of the ratio test true?

- Feb 9, 2012

- 118

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

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- Jan 26, 2012

- 268

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

I wonder if there are any non-trivial counterexamples.

- Jan 29, 2012

- 1,151

The "ratio test" says that if $lim \frac{a_{n+1}}{a_n}< 1$ then $\sum a_n$ converges.0+0+0+... converges but the ratio is not defined.

I wonder if there are any non-trivial counterexamples.

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.

- Jan 31, 2012

- 54

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

I wonder if there are any non-trivial counterexamples.

Maybe...

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

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- Jan 26, 2012

- 268

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

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