I understand the concept of supremum and infimum for monotone increasing or decreasing sequence, but this one is different. I know that for monotone increasing/decreasing sequences, you should find the upperbound and lowerbound first, and the process is usually straightforward.
I need help to find supremum and infimum of the sequence
$$(-1)^n + \frac{(-1)^{n+1}}{n}$$for $n=1$ to infinity.
I tried calculating the first 5 terms of this sequence, which are 0, 1/2, -2/3, 3/4, and -4/5. From first glance, I cannot determine the upper or lower bounds. I do have an idea that I could perhaps separate the negative and positive terms, so {1/2, 3/4,...} and {0, -2/3, -4/5,...} From this, I can see that 1/2 is the lowerbound for the positive term, and 0 is the upperbound for the negative term.
If $n$ is even, say $n=2k\;(k\geqslant 1)$, $$a_{2k} \;=\;(-1)^{2k}+\frac{(-1)^{2k+1}}{2k}\;=\;1-\frac1n\;>\;0.$$
If $n$ is odd, say $n=2k-1\;(k\geqslant 1)$, $$a_{2k-1} \;=\;-1+\frac1n\;\leqslant\;0.$$
So, for all $k$ and for all $m$, we have $a_{2k}>a_{2m-1}.$ Thus to find the overall supremum, you need only find the supremum of what you call the positive terms.
Since you understand the concept of supremum for a monotone increasing sequence, I leave the rest to you.