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?
 A: $$\sum_{n=0}^{7}|(-1)^{[n/4]}+(-1)^{[n/2]}+(-1)^{n}|$$
where $[x]$ is the integer that satisfies $[x]\leq x < [x]+1$.
A: $$\sum_{a=0}^1 \sum_{b=0}^1 \sum_{c=0}^1 |(-1)^a + (-1)^b + (-1)^c|$$
A: $\sum_{i,j,k=0}^1 | (-1)^i + (-1)^j + (-1)^k |$
A: $$
\sum_{X\in\{-1, +1\}^3}  \vert \sum_{i = 0}^{2} X_i \vert
$$
A: A less formal, but common style, would be
$$ \sum_{\epsilon_i = \pm 1} \lvert \epsilon_1+\epsilon_2+\epsilon_3 \rvert $$
A: $$\sum_{i,\ j,\ k\ =\ -1}^1 |ijk\left(i + j + k\right)|$$
$ijk$ vanishes when any of them are zero, allowing the summation to be written in the standard form.
A: 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.
