Trying to arrange sums in neat way

Question

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?

Answer 1

If I'm not mistaken, the following should be an equivalent expression:

$$\forall i \in \{1,2,3,4\}: \sum_{k=1}^{5-i} \frac{\sum_{j=k}^{k+i-1} y_j}{1+\sum_{j=k+i}^{5} y_j} \leq 1.$$

It is obviously way less clear and intuitive than the original, though.

Answer 2

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.