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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:

$$\sum \limits_{l=1}^{L}a_l^2 + \sum \limits_{l=1}^{L}\sum \limits_{l\neq j}^{L}a_l^2 \frac{a_j}{a_l}$$

Could this be re-written as:

$$\sum \limits_{l=1}^{L}a_l^2 \left(1 + \sum \limits_{l\neq j}^{L}\frac{a_j}{a_l}\right)$$

?

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?

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 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:

$$\sum \limits_{l=1}^{L}a_l^2 + \sum \limits_{l=1}^{L}\sum \limits_{l\neq j}^{L}a_l^2 \frac{a_j}{a_l}$$

Could this be re-written as:

$$\sum \limits_{l=1}^{L}a_l^2 \left(1 + \sum \limits_{l\neq j}^{L}\frac{a_j}{a_l}\right)$$

?

I have this expression,

$$\sum \limits_{l<j}^{L}a_l^2 \frac{a_j}{a_l}$$

and I would like to take $$\sum \limits_{l<j}^{L}a_l^2$$ out. That is:

$$\sum \limits_{l=1}^{L}a_l^2 \sum \limits_{l<j}^{L}\frac{a_j}{a_l}$$

I know that as it is, this is wrong. Do you have any suggestion?

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?