I am looking at trying to justify an example to myself (I'm not trying to prove anything I just want to have an idea of what is going on). If I consider $r\in\mathbb{R}, r\neq 0$ what can we say about the following:

$\lim\inf\limits_{n\to\infty} (-1)^n(r^n-r^{-n})$

$\lim\sup\limits_{n\to\infty} (-1)^n(r^n-r^{-n})$

$\lim\limits_{n\to\infty} (-1)^n(r^n-r^{-n})$

So for obvious cases, like $r=1$ this is a constant sequence so it converges and theyre all the same. For example, if $0<r<1$ it seems this would diverge, but can we say anything for the lim inf and lim sup? I'm just trying to get a handle on cases where there is no limit how to justify intuitively what to look for.


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