Timeline for Limit of empty sum
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Jun 27, 2021 at 20:36 | comment | added | Daniel Lichtblau | Empty sums are explained here. More generally, when there is an identity element for a given algebraic structure and operation, that element is taken as the result of performing the operation with no operands. | |
| Jun 25, 2021 at 19:29 | comment | added | bRost03 | I'll admit it is less clear cut than I was thinking and it depends on how you define a sum. Wikipedia defines a finite sum by $\sum_{i=a}^b g(i)=0$, for $b < a$ and $\sum_{i=a}^b g(i)=g(b)+\sum_{i=a}^{b-1} g(i)$, for $b ≥ a$ (en.wikipedia.org/wiki/Summation#Formal_definition). However, in the same article they also say (5th identity in General identities section): $\sum_{n=s}^t f(n) =\sum_{n=s}^j f(n) + \sum_{n=j+1}^t f(n)$ which contradicts their own definition. So I'll concede it's not obvious and the documentation should make clear which convention they use. | |
| Jun 25, 2021 at 18:56 | comment | added | user64494 | Sorry, your statement " it's fairly common knowledge that the empty sum is 0 and follows from basic properties of sums." is not grounded. I don't find it in the documentation. Can you give a reference? TIA. | |
| Jun 25, 2021 at 18:48 | comment | added | bRost03 |
I'm not sure how my words were emotional, but I legitimately don't understand the source of your confusion. Please elaborate and I'll give you a detailed argument. Do you not buy that the empty sum should be 0? Or that it's common knowledge that the empty sum is $0$? Or that $\sum_{i=a}^b f(i) \ ; \ b<a$ is empty? Or that Sum[1/i,{i,1,-1}] is empty? Or what?
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| Jun 25, 2021 at 18:43 | comment | added | user64494 | Sorry, I prefer arguments over emotional words. | |
| Jun 25, 2021 at 18:39 | comment | added | bRost03 | I assume it's not in the documentation because it's fairly common knowledge that the empty sum is $0$ and follows from basic properties of sums. It's empty because $\sum_{i=a}^b$ means $i$ takes on values $a\leq i \leq b$ and there are no values of $i$ such that $1\leq i \leq -1$. | |
| Jun 25, 2021 at 18:35 | comment | added | user64494 |
bRost03 (@ does not work.): You wrote in your answer "Sum[1/i,{i,1,-1}]$=0$". Don't hesitate to ask for further explanation in need.
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| Jun 25, 2021 at 18:30 | comment | added | bRost03 | I'm not sure what the relevance of that is? | |
| Jun 25, 2021 at 18:27 | comment | added | user64494 |
I don't find in the documentation to Sum something similar to Sum[...,{i,1,-1}].
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| Jun 25, 2021 at 18:16 | comment | added | bRost03 | I'm aware, it's the first example I gives under "Consider the following:". I was just showing the more general form - it's good to avoid hard-coding variables when possible. | |
| Jun 25, 2021 at 18:14 | comment | added | user64494 |
Sum[1/i, {i, 1, -1}] produces 0 without ant With.
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| Jun 25, 2021 at 18:10 | history | answered | bRost03 | CC BY-SA 4.0 |