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I don't know how to go about finding maxima/minima of such sets. What about a set like {$\frac{m+n}{m+2n}, n,m \in \mathbb N$}?

I'd like to treat the numerator and denominator seperately, and then say that the ratio of the max over the min is the set's max, and that the ratio of the min over the max is the min of the total set, but since the num. and den. are using the same variables, I can't really split them up, so that doesn't really help.

I'm assuming we can solve this by treating the set as a function and analyzing the derivatives/partial-derivatives, but I don't think that's how we're expected to do it, since we haven't done that yet, so I'd like to avoid that method, if it is indeed one.

Any guidance is appreciated. Thank you.

iaskdumbstuff
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  • it may help to break up the sequence into two subsequences, one for odd $n$ and one for even $n$ – operatorerror Apr 14 '18 at 21:14
  • @user494405 Please remember that you can choose an answer among the given if the OP is solved, more details here https://meta.stackexchange.com/questions/5234/how-does-accepting-an-answer-work – user Apr 19 '18 at 19:19

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HINT

Note that

  • n even $\frac{2n+(-1)^n}{n+2}=\frac{2n+1}{n+2}=\frac{2n+4-3}{n+2}=2-\frac{3}{n+2}$
  • n odd $\frac{2n+(-1)^n}{n+2}=\frac{2n-1}{n+2}=\frac{2n+4-5}{n+2}=2-\frac{5}{n+2}$
user
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  • Oh that's interesting. I hadn't looked at it that way. So can we even say that 2 is the max here? Doesn't the max have to be inside the set itself, which 2 isn't, right? On another note: Let's call my set $S$. I can safely say that the $min(S) = min(min(S_even), min(S_odd))$, correct? Thanks for your help. – iaskdumbstuff Apr 14 '18 at 21:24
  • @user494405 From here we can see that 2 is the supremum thus the set has not a maximum – user Apr 14 '18 at 21:26
  • I'm sorry, I don't see how that follows? I understand that it doesn't have a maximum, but I don't see how the fact that 2 is a supremum implies that the set doesn't have a maximum. – iaskdumbstuff Apr 14 '18 at 21:28
  • @user494405 Since $3/(n+2)$ and $5/(n+2)$ are monotonic (strictly decreasing) then $2-3/(n+2)$ and $2-5/(n+2)$are also monotonic (strictly increasing) and thus we can exclude that exist $\bar n$ such that $f(\bar n) \ge 2$. – user Apr 14 '18 at 21:34