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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:

$$\sum_{i=1}^{n} wd\sum_{i=1}^{n} wc$$

John V
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2 Answers2

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$\sum_{i=1}^{n} w_i + \sum_{i=1}^{n} v_i$ 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:

$$\left(\sum_{i=1}^{n} w_i \right)\left(\sum_{j=1}^{n} v_j\right)$$

Thomas Andrews
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  • If I may ask, why the other sigma uses "j" as index? is it necessary? In my fraction, there are three sigmas (but two are the same). The number of elements is different between the two unique ones. – John V Oct 09 '18 at 14:55
  • Not necessary.. – Lubin Oct 09 '18 at 15:03
  • It is not wrong to use the same variables. But if $i$ is say, running over the rows of an $n\times n$ matrix, and $j$ is running over the columns, then you might want to use different variables. For the product, it is nice to use different variables because the product is equal to: $$\sum_{i=1}^n\left(\sum_{j=1}^{n} w_iv_j\right)$$ which would be awkward if both were $i.$ (For the sum, it is easier to use both as $i$, but if $i,j$ are two different "types" it might not make sense to match $i=1$ with $j=1.$) Or if the sums had different ranges. – Thomas Andrews Oct 09 '18 at 15:03
  • I consider it fruitful to let $\sum_{i=1}^{n} w_i\sum_{j=1}^{n} v_j$ be sum within a sum. If $v_j$ depends on $i$, then it's necessary, and if not then it doesn't matter. So why not let it be consistent? – Arthur Oct 09 '18 at 15:21
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$$\sum_i a_i\sum_j b_j$$ 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 $+$

$$\sum_i a_i\sum_j b_j\ne\color{red}{\sum_i a_i+\sum_j b_j}.$$