Talk:Ratio test

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[edit] radius of convergence

I removed anything about the radius of convergence because it wasn't clear. This test has a similar form when used for radius of convergence but aren't the same. Fresheneesz 01:58, 29 March 2006 (UTC)

I hate to ask but is there a test that is similar to the ratio test (you state) but without the Absolute values? I only ask because in my text this is called the "Ratio test for absolute convergence." I realize this is only convention, but should there be a mention of the "ratio test of regular convergence"? -Nightwindzero 14:46, 20 April 2007 (UTC)

If a sequence or series is absolutely convergent, it is (in your case, "regularly") convergent. In other words, absolute convergence is, if you like, "stronger" than convergence. x42bn6 Talk 23:52, 20 April 2007 (UTC)

[edit] unclarity

The edits by user:Fresheneesz made the article very unclear. It said:

In mathematics, the ratio test is a test (or "criterion") for the convergence of a series whose terms are real or complex numbers. The test was first published by Jean le Rond d'Alembert and is sometimes known as d'Alembert's ratio test. The ratio test is defined as:
L = \lim_{n\rightarrow\infty}\left|\frac{a_{n+1}}{a_n}\right|
where
lim denotes the limit as n goes to infinity,
an and an+1 are the nth and (n+1)th terms of an infinite series
and
L is a label for the result of the ratio test.
The results of the ratio test show that:
  • if L<1 \! the series converges absolutely, or
  • if L>1 \! the series diverges, or
  • if L=1 \! the test is inconclusive (there exist both convergent and divergent series that satisfy this case).
For example, any series in the form:
\sum_{n=1}^\infty f_n
can be applied to the ratio test.

So it began by saying that the test is a particular number. That is nonsense.

It said "the results of the test show that" when it meant "the test states that".

It said "or" where it meant "and".

It said that a series can be applied to the test, where it meant the test can be applied to a series.

It said of a particular form, and offered this as an example, but the particular form was not a particular form, but completely general: ALL series are of that form. This does not constitute an "example". But the sentence began by saying "For example, ...".

It said "... where ... an and an+1 are the nth and (n+1)th terms of an infinite series". That infinite series was the topic of the whole account of what the test says; to relegate this to a "where..." clause that appears only AFTER the limit L is mentioned is objectionable on several levels, logical and pedagogical.

I mention all this here lest anyone consider reverting my edits.

I was led to this by a comment at talk:radius of convergence under the heading "odd wording". That led me to edit root test and then to edit the present article. Michael Hardy 22:14, 31 October 2006 (UTC)

[edit] Weaker conditions?

Isn't this ratio test's condition slightly too strong? I learnt the following as the "ratio test":

If there exists a number M < 1 such that \left|\frac{a_{n+1}}{a_n}\right| \leq M for all sufficiently large n, then \sum_{n=0}^\infty a_n converges absolutely.

So if that ratio keeps oscillating between, say, 0.1 and 0.9, I should still be able to say the series converge absolutely, even though the limit of that ratio won't exist.

Similar thing if there's some number M > 1 so that the ratio is not less than M for all sufficiently large n... -- 203.171.200.81 (talk) 13:52, 17 November 2007 (UTC)

Don't they, semantically, mean the same thing? "For sufficently large n" seems semantically equivalent to limits. x42bn6 Talk Mess 17:17, 17 November 2007 (UTC)