1
$\begingroup$

What word properly completes the phrase

the radius of convergence does not depend on the $\text{______}$ of the interval

to mean that it doesn't matter whether $(a, b)$, $[a, b)$, $(a, b]$, or $[a, b]$ is the correct answer?

  • Openness and closedness don't really seem to work because the interval doesn't have to be either (it could be half-open, or, in $\mathbb{R}^n$, include any subset of its limit points).
  • Strictness makes sense, because you can say that $2$, and not $3$, is "strictly between" $1$ and $3$. However, this only really makes sense (to me) once you know the meaning; if I saw the word strictness I wouldn't really know what it meant.
  • Boundary and endpoints don't work because the boundary does matter—we care what $a$ and $b$ are, just not whether they're included in the interval.

This is for a Calculus II class, so topology, etc. are outside the scope of the curriculum.

Thoughts?

$\endgroup$
8
  • $\begingroup$ @BrianFitzpatrick I don't think that works because it seems to imply that we don't care what $a$ and $b$ are. $\endgroup$
    – wchargin
    May 13, 2014 at 0:35
  • $\begingroup$ Maybe "half-open/closedness" then? $\endgroup$
    – Brian Fitzpatrick
    May 13, 2014 at 0:37
  • $\begingroup$ "Half-open/closedness" is close but a bit too specific to $\mathbb{R}^1$. This is for a Calculus II class, so "topology" wouldn't be understandable. $\endgroup$
    – wchargin
    May 13, 2014 at 0:40
  • 1
    $\begingroup$ Yes but your context is the interval of convergence of a power series. There's no reason to consider anything other than $\Bbb R^1$. $\endgroup$
    – Brian Fitzpatrick
    May 13, 2014 at 0:47
  • 2
    $\begingroup$ I might use the word 'type', or I would just rewrite the sentence (the radius of convergence does not depend on whether the interval includes the endpoints or not). $\endgroup$
    – Alex Zorn
    May 13, 2014 at 0:53

2 Answers 2

Reset to default
1
$\begingroup$

I would use:

the radius of convergence does not depend on the nature of the interval, i.e.,whether it is closed, open, or neither.

$\endgroup$
0
$\begingroup$

"The radius of convergence does not depend on whether the interval is open, closed, or neither."

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .