# summa

• December 29th 2007, 10:42 PM
DivideBy0
summa
I've encountered lots of other 'ways' of expressing a sum that are different to

$\sum_{k=1}^n$

Sometimes people put $\sum_{cyc}$, $\sum_{sym}$, $\sum_{a,b,c}$, and sometimes they just put $\sum$. What does this all mean?? Thanks!
• December 29th 2007, 10:53 PM
Isomorphism
Quote:

Originally Posted by DivideBy0
I've encountered lots of other 'ways' of expressing a sum that are different to

$\sum_{k=1}^n$

Sometimes people put $\sum_{cyc}$, $\sum_{sym}$, $\sum_{a,b,c}$, and sometimes they just put $\sum$. What does this all mean?? Thanks!

I will write for the three variable case:
cyc means for all "cyclic" permutations
$\sum_{cyc}f(x,y,z) =f(x,y,z)+f(y,z,x)+f(z,x,y)$

I am not sure about this, but I think this means $\sum_{sym}$ means all possible permutations.
Meaning for three variable case, there will be 6 terms!
• December 30th 2007, 07:51 AM
ThePerfectHacker
Quote:

Originally Posted by DivideBy0
I've encountered lots of other 'ways' of expressing a sum that are different to

$\sum_{k=1}^n$

Sometimes people put $\sum_{cyc}$, $\sum_{sym}$, $\sum_{a,b,c}$, and sometimes they just put $\sum$. What does this all mean?? Thanks!

Here is an example. As we want to prove that $\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b} \geq \frac{3}{2}$ for positive numbers.
We can write,
$\sum_{cyc}\frac{a}{b+c} \geq \frac{3}{2}$
And people take that to mean the entire cyclic sum.
• December 30th 2007, 09:38 PM
Isomorphism
Hey DivideBy0,
Just thought I will add an important point in reading these sums
Can you tell me what is $\sum_{sym}a$?
Consider I have only 3 variables a,b,c...

Be careful it is a little tricky :)
• January 6th 2008, 12:11 PM
DivideBy0
Quote:

Originally Posted by Isomorphism
Hey DivideBy0,
Just thought I will add an important point in reading these sums
Can you tell me what is $\sum_{sym}a$?
Consider I have only 3 variables a,b,c...

Be careful it is a little tricky :)

Well... I think that

$\sum_{sym}a=a+b+c$ ... :) hope it's right

Also, there is a section on this in this link.

It is in 'A less than B' Chapter 2.2

http://www.mathhelpforum.com/math-he...ss-than-b.html

Thanks PaulRS for showing me the link, and of course Krizalid and ThePerfectHacker for posting the documents.