$\inf$ and $\sup$ of a set given by $\sum\limits_{k=1}^{n}\frac{a_{k}}{a_{k}+a_{k+1}+a_{k+2}}$

Your bounds are sharp (to my surprise).

Example: $a_k = x^k$. Then \begin{align*} \sum_{k=1}^n \frac{a_k}{a_k + a_{k+1} + a_{k+2}} &= \sum_{k=1}^{n-2} \frac{x^k}{x^k + x^{k+1} + x^{k+2}} + \frac{x^{n-1}}{x^{n-1} + x^n + x} + \frac{x^n}{x^n + x + x^2} \\ &= \frac{n-2}{1 + x + x^2} + \frac{1}{1 + x + x^{-n+1}} + \frac{1}{1 + x^{-n+1} + x^{-n+2}} \end{align*} This tends to $1$ as $x\to\infty$ and to $n-2$ as $x\to 0^+$.

The only essential feature of the functions $x^k$ here is that each grows faster than the previous one (as $x\to\infty$) and shrinks faster than the previous one (as $x\to 0^+$).