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

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Discussion Overview

The discussion centers around the validity of the converse of the ratio test for series convergence. Participants explore whether a convergent series necessarily implies that the limit of the ratio of consecutive terms is less than 1, examining both trivial and non-trivial examples.

Discussion Character

  • Debate/contested

Main Points Raised

  • Some participants express skepticism about the truth of the converse of the ratio test, suggesting that counterexamples can be constructed.
  • One participant notes that the series \(0 + 0 + 0 + \ldots\) converges, but the ratio is not defined, raising questions about non-trivial counterexamples.
  • Another participant proposes the series \(a_n = \frac{1}{n(n-1)}\) as a potential non-trivial counterexample.
  • There is a suggestion to find a convergent series where the limit of the ratio is equal to 1, indicating that the limit may not always be less than 1 for convergent series.
  • A participant mentions the series \(\sum \frac{1}{n^2}\) in the context of discussing limits of ratios.
  • One participant questions whether it is safe to conclude that if a series converges, then \(\lim\left|\frac{a_{n+1}}{a_n}\right| \le 1\).

Areas of Agreement / Disagreement

Participants generally disagree on the validity of the converse of the ratio test, with multiple competing views and examples being presented. The discussion remains unresolved regarding the existence of definitive counterexamples.

Contextual Notes

Participants have not reached a consensus on the implications of the ratio test's converse, and there are unresolved questions regarding the definitions and conditions under which the limit of the ratio is evaluated.

alexmahone
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Is the converse of the ratio test true?
 
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I don't think so. I think you can construct an easy counterexample. Care to imagine one?
 
Krizalid said:
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.
 
Alexmahone said:
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.
 
Alexmahone said:
0+0+0+... converges but the ratio is not defined.

I wonder if there are any non-trivial counterexamples.
Maybe...

$$ a_n=\frac{1}{n(n-1)} $$
 
HallsofIvy said:
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 $\displaystyle\lim\left|\frac{a_{n+1}}{a_n}\right|\le 1$?
 

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