How do you find the maximum of a set like {$\frac{2n+(-1)^n}{n+2}, n \in \mathbb N$}?

Question

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.

Answer 1

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 $\frac{2n+(-1)^n}{n+2}=\frac{2n-1}{n+2}=\frac{2n+4-5}{n+2}=2-\frac{5}{n+2}$