TITLE: Trying to arrange sums in neat way
QUESTION [7 upvotes]: I am trying to write the following summations as compactly as possible using summation and enumeration:

$$\frac{y_1}{1+y_2+y_3+y_4+y_5}+\frac{y_2}{1+y_3+y_4+y_5}+\frac{y_3}{1+y_4+y_5}+\frac{y_4}{1+y_5} \leq 1$$
$$\frac{y_1+y_2}{1+y_3+y_4+y_5}+\frac{y_2+y_3}{1+y_4+y_5}+\frac{y_3+y_4}{1+y_5} \leq 1$$
$$\frac{y_1+y_2+y_3}{1+y_4+y_5}+\frac{y_2+y_3+y_4}{1+y_5}\leq 1$$
$$\frac {y_1+y_2+y_3+y_4}{1+y_5}\leq 1$$

I wrote it as following:
$$\sum_{k=1}^4 \frac {y_k}{1+y_{k+1}+...+y_5} \leq 1$$
$$\sum_{k=1}^3 \frac {y_k+y_{k+1}}{1+y_{k+2}+...+y_5} \leq 1$$
$$ \sum_{k=1}^2 \frac {y_k+y_{k+1}+y_{k+2}}{1+y_{k+3}+...+y_5} \leq 1$$
$$ \frac {\sum_{k=1}^4 y_k}{1+y_5} \leq 1$$

Is there any enumeration to combine the $4$ summations?

REPLY [3 votes]: What you gain it compactness you lose (massively) in comprehsion but:
$\frac {y_1}{1 + y_2 + ... + y_5} + .... = \sum_{k=1}^4 \frac {y_k}{1+\sum_{j=k+1}^5 y_j}\le 1$
And $\frac{y_1+y_2}{1+y_3+y_4+y_5}+\frac{y_2+y_3}{1+y_4+y_5}+\frac{y_3+y_4}{1+y_5} = \sum_{k=1}^{5-1}\frac {y_k + y_{k+1}}{1+\sum_{j=k+2}^5 y_j}\le$
And then you get
$[\sum_{k=1}^{5-m}\frac {\sum_{i=k}^{k+m-1} y_i}{1+\sum_{j=k+m}^5 y_j}  \le 1]|_{m=1}^4$
But really, if a were a reader  and I came across that... I'd hit you.