OPTION 1.
I have this expression,
$$\sum \limits_{l=1}^{L}a_l^2 + \sum \limits_{l<j}^{L}a_l^2 \frac{a_j}{a_l}$$
and I would like to take out $$\sum \limits_{l=1}^{L}a_l^2$$, as to obtain something like:
$$\sum \limits_{l=1}^{L}a_l^2 \left(1 + \sum \limits_{l<j}^{L}\frac{a_j}{a_l}\right)$$
Does it make sense? Do you have any suggestion?
OPTION 2.
Alternatively, the expression above can also be re-written in a different way:
$$\sum \limits_{l=1}^{L}a_l^2 + \sum \limits_{l\neq j}^{L}a_l^2 \frac{a_j}{a_l}$$
which should be equivalent to (following Double summation index notation: $\Sigma_{i<j}$ versus $\Sigma_{i\neq j}$?):
$$\sum \limits_{l=1}^{L}a_l^2 + \sum \limits_{l=1}^{L}\sum \limits_{l\neq j}^{L}a_l^2 \frac{a_j}{a_l}$$
Because of the distributivity property of summations (https://en.wikipedia.org/wiki/Summation), such as:
$$\left(\sum \limits_{l=1}^{L}a_l^2\right) \left(\sum \limits_{l\neq j}^{L}\frac{a_j}{a_l}\right)=\sum \limits_{l=1}^{L}\sum \limits_{l\neq j}^{L}a_l^2 \frac{a_j}{a_l}$$
we could re-write the original expression as:
$$\sum \limits_{l=1}^{L}a_l^2 \left(1 + \sum \limits_{l\neq j}^{L}\frac{a_j}{a_l}\right)$$
Does it seem correct to you?
