Talk:Radius of Convergence from Limit of Sequence/Real Case

$1/R = 0$

The condition $1/R = 0$ as stated in the Theorem is, of course, gibberish mathematically. (It seems odd that there is a comment at the bottom saying that the alternative definition of R would be "less convenient".) However, I assume that this was copied from the original source. Are we therefore required to keep it as is? KarlFrei (talk) 09:55, 14 November 2018 (EST)

Page 144 of Binmore (cited on this page) gives it like this, but qualifies with "appropriate conventions must be adopted if the right hand side of the formula happens to be $0$ or $+\infty$. I think it is easily rectified by just writing $\ds \lim_{n \mathop \to \infty} \size {\frac {a_{n + 1} } {a_n} } = 0$ without the $\dfrac 1 R$ as in the Cauchy-Hadamard Theorem, and writing some justification in the body of the proof. Caliburn (talk) 11:42, 14 November 2018 (EST)

Thanks. I still don't understand why we don't simply write $\ds R = \lim_{n \mathop \to \infty} \size {\frac {a_{n - 1} } {a_n} }$ though (possibly with a comment that if $\ds \lim_{n \mathop \to \infty} \size {\frac {a_{n - 1} } {a_n} } = \infty$, the interval of convergence is $\R$). In what way is this less convenient?

Also, the current theorem elides the fact that the limit might not exist. I will add this comment since Binmore also makes it (phew!). KarlFrei (talk) 03:46, 15 November 2018 (EST)

This is consistent with the form given in Ratio Test (and in fact can be deduced directly from this result) and the Cauchy-Hadamard Theorem. I do know that some people do not like writing $\ds \lim \cdot = \infty$, but it is unambiguous written as shorthand. (I have always written $\cdot \to \infty$ as $\cdots$ here) Having skimmed some other pages on the topic, $1/R = \ldots$ seems to be the common convention, so I'd leave it as is. That said I'd have no issue writing $\ds R = \lim_{n \mathop \to \infty} \size {\frac {a_{n - 1} } {a_n} }$ or equivalent given appropriate clarification as you say. Caliburn (talk) 04:37, 15 November 2018 (EST)