How would I express the series $|1+1+1|+|1+1-1|+|1-1+1|+|1-1-1|+|-1+1+1|+|-1+1-1|+|-1-1+1|+|-1-1-1|$ in summation notation?

I tried putting the series $$ |1+1+1|+|1+1-1|\\+|1-1+1|+|1-1-1|\\+|-1+1+1|+|-1+1-1|\\+|-1-1+1|+|-1-1-1| $$ into Wolfram Alpha and typing "in summation notation" but it wouldn't tell me what it is in summation notation. I tried to figure it out on my own but I can't figure out how to put this series into summation notation.

How would I express this sequence in summation notation?

Solution 1:

$\sum_{i,j,k=0}^1 | (-1)^i + (-1)^j + (-1)^k |$

Solution 2:

A less formal, but common style, would be $$ \sum_{\epsilon_i = \pm 1} \lvert \epsilon_1+\epsilon_2+\epsilon_3 \rvert $$

Solution 3:

There are several options. It's good to know many; different ideas are convenient in different situations.

Here is one that hasn't been suggested yet but I would recommend considering: $$ \sum_{a=\pm1}\sum_{b=\pm1}\sum_{c=\pm1}|a+b+c|. $$ This is quite similar to the one by Chappers, but the three sums are more explicit here. You can also consider replacing "$a=\pm1$" with "$a\in\{-1,+1\}$".

Here are some more options for the record: $$ \sum_{a,b,c=\pm1}|a+b+c|,\\ \sum_{a,b,c=-1,+1}|a+b+c|,\\ \sum_{a,b,c\in\{-1,+1\}}|a+b+c|,\\ \sum_{a=-1,+1}\sum_{b=-1,+1}\sum_{c=-1,+1}|a+b+c|. $$ As roger suggested in another answer, you can also let each index $i$ go from 0 to 1 and then sum $(-1)^i$. This leads to many more variations of the formulas above.