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In Rudin, principles of mathematical analysis https://web.math.ucsb.edu/~agboola/teaching/2021/winter/122A/rudin.pdf p.68, he claims that whenever the root test is inconclusive then the ratio test is aswell. But lets consider the sequence

$1 + 1 + ...$

Clearly $\left|\frac{a_{n+1}}{a_n}\right| = 1 \quad \forall n$ therefore the ratio test implies divergence but $$\limsup_{n \to \infty} |a_n|^{1/n} = 1$$ so the root test gives no information. Is this a counter example to Rudin's claim or am I missing something?

Edit: It seems that there is cofusion here around the definition of the ratio test. Rudin claims (p.66) that a sequence diverges if $$\exists n_0 \text{ such that } n>n_0\text{ implies } \left|\frac{a_{n+1}}{a_n}\right| \geq 1.$$ There may be other definitions of the ratio test but with this definition is the example given a counter example?

G Aker
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    If $a_n=\frac1{n^2}$, then we also have $\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|=1$. Are you claiming that the series $\sum_{n=1}^\infty\frac1{n^2}$ diverges? – Another User Feb 02 '23 at 09:10
  • @AnotherUser that sequence doesn't satisfy the ratio test as given in the reference, whereas my sequence does. See the edit above. – G Aker Feb 02 '23 at 10:19
  • (after your Edit) Unless I'm missing something, I think you're correct. What Rudin specifically labels as the ratio test (p. 66) makes the p. 68 comment not fully correct. The comment pertains to the ratio and root tests stated in terms of limiting behavior, which is slightly weaker than the divergence criteria he gave in the ratio test theorem. I suspect that after improving the preciseness of the divergence half of the theorem, he apparently forgot that the usual statement one makes about the ratio and root tests (his p. 68 comment) needs adjustment, which is strange for a 3rd edition. – Dave L. Renfro Feb 02 '23 at 12:14

2 Answers2

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The ratio test is inconclusive in the case where $$\left|\frac{a_{n+1}}{a_n}\right| = 1.$$ There are both convergent and divergent series for which this ratio equals $1$. The link illustrates examples of such cases. Therefore, your example does not furnish a counterexample to the claim.

heropup
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The ratio test giving you a $1$ is not divergence, but rather says that it is inconclusive. Also the ratio test looks at the limit, not simply the ratio for a fixed $n$.

In fact your example is one of the examples on the Wikipedia page.

Wikipedia seems to say that the root test actually implies divergence if the $\limsup$ is $1$ and it approaches strictly from above; I wasn't aware of this, I'm pretty sure Rudin doesn't mention this form of the root test either, but if this is true, this actually shows that your example diverges.

The root test is stronger than the ratio test for the simple reason that the limit of the expression in the root test is less than or equal to that of the ratio test; so if the limit exists in the latter, the limit must exist in the former. This answer gives a formal proof.

Boshu
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  • See the edit in the question, I am referring specifically to the form of the ration test given by Rudin. – G Aker Feb 02 '23 at 10:21
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    @GAker I see what you mean; I think the point you make is fair, but I think the comment on page 68 in isolation should be taken to be meant about the standard form of the ratio test. A comparison like the ones Rudin mentions is only to make the point that I made in the last paragraph of my answer, so it makes sense to compare equivalent conditions. The additional condition in the root test that I mention from Wikipedia is actually the root test counterpart of the part (b) of Thm 3.34 (b); that will also give you same conclusion. – Boshu Feb 02 '23 at 10:54
  • @Boshu: I didn't see your comment until after I made mine above (the one made after the Edit), but I definitely feel better after seeing it just now. I find it hard to believe that a book as well used as Rudin's (even in the 1950s and 1960s, not just at present) has something like this in its 1976 3rd edition, so I was really expecting to later find that someone would point out something that I was overlooking in my comment. – Dave L. Renfro Feb 02 '23 at 12:19
  • @DaveL.Renfro I agree, I'm also quite surprised. I haven't looked at Baby Rudin in maybe six or seven years, but I do not recall this statement at all although it is quite simple in hindsight. Maybe we had a different edition, but it is rather surprising that the ratio test included the stronger form but not the root test. I'd only studied the limiting behavior as an undergrad if I recall correctly. – Boshu Feb 02 '23 at 19:05