I have a fraction where in the denominator I need to sum two products (numbers and their weights). For this, I inserted two Sigmas but now I am unsure whether I can have Sigma + Sigma or just Sigma Sigma to denote sum of their respective results.
$$\sum_{i=1}^{n} wd+\sum_{i=1}^{n} wc$$
The first is a sum of Ws and Ds, the other is a sum of Ws and Cs. Somewhere I have seen the notation withou the plus sign, just one next to the other, like this:
$$\sum_{i=1}^{n} wd\sum_{i=1}^{n} wc$$
On
$$\sum_i a_i\sum_j b_j$$ can be interpreted as the product
$$\left(\sum_i a_i\right)\left(\sum_j b_j\right).$$
And
$$\sum_i a_i\sum_j b_{ij}$$ has only meaning as a sum of products
$$\sum_i\left(a_i\sum_j b_{ij}\right).$$
In these cases, there is no ambiguity with
$$\sum_i\sum_j a_i b_j$$ nor
$$\sum_i\sum_j a_i b_{ij}.$$
But there is never an implied $+$
$$\sum_i a_i\sum_j b_j\ne\color{red}{\sum_i a_i+\sum_j b_j}.$$
$\sum_{i=1}^{n} w_i + \sum_{i=1}^{n} v_i$ is the some of the two sums. $\sum_{i=1}^{n} w_i\sum_{i=1}^{n} v_i$ is the product of the two sums.
Sometimes you'll want to clarify that the second sigma is not "inside" the first sigma, so you'll write:
$$\left(\sum_{i=1}^{n} w_i \right)+ \left(\sum_{j=1}^{n} v_j\right)$$
and:
$$\left(\sum_{i=1}^{n} w_i \right)\left(\sum_{j=1}^{n} v_j\right)$$