What's the supremum of the following set $\{ n + \frac{(-1)^n}{n} : n \in \mathbb{N}\}$ What's the supremum of the following set $\{ n + \frac{(-1)^n}{n} : n \in \mathbb{N}\}$?
I know that the infimum is $0$, but what about the supremum? I have calculated with Maxima the first $1000$ terms, and it seems that the numerator grows faster than the denominator (because we are adding or removing always a smaller fraction), could it be $\infty$?

To see that it isn't bounded above, observe that for a positive integer $k$
$$2k + \frac{(-1)^{2k}}{2k} = 2k + \frac1{2k}>2k. $$
Given $M>0$, choose $k$ such that $2k>M$. Then $2k + \frac1{2k}$ is an element of the bracketed set which is greater than $M$.
