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If I were to simplify a summation that contains a formula that is not a constant and does not contain the index, what would it simplify down to?

For example, if were to simplify the following summation:

$$ \sum_{i=1}^{100} j $$

Would the result be equal to 100j or 0?

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  • $\begingroup$ it is constant with respect to the summation index $i$. But strictly speaking you should write $\sum_{i=0}^{100} j$ or $\sum_{I=1}^{100} j$ or $\sum_{I=47}^{100} j$ or wtv, as it is its not clear. If it is $\sum_{I=A}^{100} j$ then it is $$\sum_{I=A}^{100} j = j \sum_{I=A}^{100} 1 = j(100 - A + 1)$$ $\endgroup$ Commented Aug 12, 2023 at 13:34
  • $\begingroup$ What do you feel $\sum_i^{100}1$ should be? $\endgroup$ Commented Aug 12, 2023 at 13:35
  • $\begingroup$ I made an edit, I was supposed to say that the index i starts at 1 $\endgroup$ Commented Aug 12, 2023 at 13:36

1 Answer 1

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We have

$$\sum_{i=1}^{100}a_i=a_1+a_2+\cdots+a_{100}$$

In your example, $a_i=j$ for every $i$, and so

$$\sum_{i=1}^{100}a_i=\underbrace{j+j+\cdots+j}_{\text{100 times}}=100j$$

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    $\begingroup$ That makes sense, and I thought this would be the case as well. It's just that, I used some online calculators to verify my idea, and they evaluated it to 0, not 100j. $\endgroup$ Commented Aug 12, 2023 at 13:40
  • $\begingroup$ I found one such calculator that does as you say. However, if you put in a particular constant (e.g. $j=1$), then it will give what you would expect. Probably it is just that the calculator is only designed to work with expressions involving the index of summation (and not letters denoting constants). $\endgroup$
    – smcc
    Commented Aug 12, 2023 at 13:45

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