# What does the math notation $\sum$ mean?

I have come across this symbol a few times, and I am not sure what it "does" or what it means: $\Large\sum$

• Usually it is used for a sum. See this Wikipedia entry for summation – Adrián Barquero Apr 11 '11 at 1:06
• Symbolic computation, while sharing the same first word as your question does not quite relate, retagged to notation. – crasic Apr 11 '11 at 1:13
• Since nobody has mentioned it, this is the capital Greek letter, sigma, which is the Greek equivalent of the letter S. As others have noted, it is used to represent sums. – Thomas Andrews Apr 11 '11 at 1:19

Here I use it once to explain what it does.

$$\sum_{i=1}^{5} i=1+2+3+4+5$$

Which translates to, sum over $i$, where $i$ starts at $1$ and goes to $5$. or this case

$$\sum_{i=1}^{5} i^2=1^2+2^2+3^2+4^2+5^2$$

Which translates to sum over the values of $i$, which range from $1$ to $5$ the function $i^2$.

Naturally one may wonder what if it is a product we are after, for example how do I represent $1\times2\times3\times4\times5$ or $1^2\times2^2\times3^2\times4^2\times5^2$

The notation for those are

$$\prod_{i=1}^5 i$$

and

$$\prod_{i=1}^5 i^2$$

• Just curious, as a high scool student, what is that Greek letter called? – Hidde Dec 23 '11 at 15:26
• @Hidde: $\Sigma$ is the upper-case Greek letter sigma (lower case: $\sigma$). $\Pi$ is the upper-case Greek letter pi (which you may know better in lower case, $\pi$). – Nate Eldredge Dec 23 '11 at 15:43
• Thanks! Kind of logical now that I think of it, P for Product, just as S for Sum :). – Hidde Dec 23 '11 at 20:42

Coming from a programming background, I found it quite helpful to explain it using a for loop:

The mathematician would write it like this:

$\sum\limits_{i=m}^n f(i)$

And the programmer would write it like this:

result = 0
for (i=m; i<=n; i++) {
result += f(i)
}


You can think of m as the start index and n as the end index.