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What does the math notation $\sum$ mean?

My school's prescribed book uses the weird letter E character without explaining what it is in the first chapter when it talks about the binomial equation. I can't find it on Google either because I don't know what it means or its name. Please help me!

$$ (x+a)^n = \sum_{k=0}^{n} \binom{n}{k}x^ka^{n-k}$$

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marked as duplicate by t.b., robjohn, Srivatsan, J. M., Asaf Karagila Dec 23 '11 at 14:26

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It's the Greek capital letter $\Sigma$ sigma. Roughly equivalent to our 'S'. It stands for 'sum'. Read this for starters. –  Jyrki Lahtonen Nov 14 '11 at 7:27
en.wikipedia.org/wiki/Summation –  pedja Nov 14 '11 at 7:28
...and maybe it wouldn't hurt to mention that sigma is a consonant, since called it a "weird E letter" could create a different impression. –  Michael Hardy Nov 14 '11 at 12:04
In general $\displaystyle{\sum_{n=i}^{j} f(n) = f(i) + f(i+1) + \ldots + f(j-1) + f(j)}$ –  badp Nov 14 '11 at 14:12
I died a little reading this :( –  Mariano Suárez-Alvarez Jan 10 '12 at 4:41

2 Answers 2

up vote 19 down vote accepted

This is a capital sigma. Its use is best illustrated by an example:

$$ \sum_{k = 1}^4 \frac{1}{k} = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4}. $$

You begin by replacing the index (in this case, $k$) with the first value it takes on (in this case, 1). You then proceed to the next number and keep doing this replacement until you are at the upper limit (in this case, 4). Finally, you add all these terms up.

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Thanks so much for the simple explanation ! –  tina nyaa Nov 14 '11 at 7:58

As explained by Austin Mohr, it is the summation operation $\sum\limits_{k=m}^n f(k)$ (Greek uppercase letter Sigma) which sums every value of $f(k)$ where every value between $m$ and $n$ inclusively is substituted into $k$. That is:

$$\sum_{k=1}^5\frac{1}{k} = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} \approx{2.28}$$

A related operation is the product operator denoted $\prod\limits_{k=m}^n f(k)$ (Greek uppercase letter Pi) which returns the product of the terms with $k$ substituted for every value between $m$ and $n$ inclusive.

$$\prod_{k=1}^3 (k + x) = (1 + x) (2 + x) (3 + x) = x^3 + 5x^2 + 11x + 6$$

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Notice that if you write \sum instead of \Sigma in TeX, then the software will follow certain standard conventions. One of those is that when the expression is in a "displayed" as opposed to "inline" setting, the subscript and superscript appear directly below and above the Sigma. Similarly with \prod rather than \Pi. I've taken the liberty of editing this answer accordingly. –  Michael Hardy Nov 14 '11 at 12:03
@MichaelHardy Thanks Michael. Brain really wasn't fully engaged when I was typing that out it seems! –  Edd Nov 14 '11 at 14:42

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