# Supremum and infimum of $\sum_{k=1}^{n}\frac{a_k}{a_{k}+a_{k+1}+a_{k+2}}$

Let, $$n\ge 3$$ be a fixed integer. Take all the possible finite sequences $$(a_1,a_2,..,a_n)$$ of positive numbers. Find, supremum and infimum of $$\sum_{k=1}^{n}\frac{a_k}{a_{k}+a_{k+1}+a_{k+2}}$$ where, $$a_{n+1}=a_1,a_{n+2}=a_2$$

My partial progress: Let, $$s=a_1+a_2+..+a_n.$$ Since, $$\frac{a_k}{s}\le\frac{a_k}{a_k+a_{k+1}+a_{k+2}}\le 1-\frac{a_k}{s}-\frac{a_{k+1}}{s}$$ which gives, $$1\le\sum_{k=1}^{n}\frac{a_k}{a_k+a_{k+1}+a_{k+2}}\le n-2$$ I guess, $$n-2$$ and $$1$$ will be the supremum and infinmum respectively,but I failed to show that.

• What is $s$ in what you wrote? Commented Sep 10, 2018 at 2:31
• Also, your sum should stop at $n-2$ in the statement. Commented Sep 10, 2018 at 2:36
• I edited it.See now.
– SOUL
Commented Sep 10, 2018 at 2:44
• The fact that the sequence is "cyclic" ($a_{n+1}=a_1$) changes a lot of things. Commented Sep 10, 2018 at 2:48
• Commented Jun 26, 2019 at 10:53

Put $a_k=m^k$, when $m\rightarrow \infty$ the sum approaches $1$.
Put $a_k=m^{-k}$, when $m\rightarrow \infty$ the sum approaches $n-2$.