The *summation sign* (pronounced sigma) $\sum$ is used as follows:

Given any function $f(x)$ \[\sum\limits_{x=1}^{n}f(x)=f(1)+f(2)+\ldots + f(n-1)+f(n).\]

It calculates the sum of the function $f(x)$ from $x=1$ to $x=n$ (this is dictated by what is above and below the $\sum$ sign).

\begin{align*} \sum\limits_{x=1}^{5}x &= 1+2+3+4+5 = 15\text{,} \\ \sum\limits_{x=1}^{n}x &= 1+2+ \ldots + (n-1) +n. \end{align*} This time, $x$ will stay as a variable and $k$ will change from $0$ to $n$:

\[\sum\limits_{k=0}^n x^k = x^0+x^1+ \ldots + x^{n-1} + x^{n} .\]

Now a numerical example: \[\sum\limits_{k=0}^4 2^k = 2^0+ 2^1+2^2+2^3+2^4 = 31 .\] Now, if we have the constants (numbers that don't change, i.e. are constant), $c_0, c_1, \ldots , c_n$, we can write the formula for a polynomial as follows: \[\sum\limits_{i=0}^{n} c_i x^i = c_0x^0+c_1x^1+ \ldots + c_{n-1} x^{n-1} + c_nx^n .\]

**Note:** When the meaning is clear, the running variable may be dropped, giving \[\sum{ f(x)} .\]

A *set* is a collection of objects which are known as elements. The empty set, the set which does not contain any elements, is denoted by $\phi$.

We usually denote the elements in a set between curly brackets. These are a few examples of sets:

- $\{ 1, 2, 3, 4, 5\}$,
- $\{ 2, 4, 6, 8, 10 \}$,
- $\{$red, yellow, green, blue$\}$,
- $\{ 7, \pi,$ carrot, bucket, $7 \}$,
- $\mathbb{N} = \{1, 2, \ldots \} $,

**Note:** 1= This last set is called *the natural numbers*. $0$ is not usually included in this set.

- $\mathbb{R} = \{\text{any real number}\}$.

A *subset* $T$ of a set $S$ is a set which does not contain any elements that are not also contained in $S$.

We write $T \subseteq S$ to mean $T$ is a subset of $S$.

We write $T \subset S$ to mean $T$ is a strict subset of $S$, i.e. it is not the exact same set as $S$.

Now we can use sets in Sigma notation. Let $S$ be a set, $f(x)$ be a function. Then \[\sum\limits_{x\in S}f(x)\] is the sum of $f(x)$ for all values $x$ in the set $S$.

Consider the sets $R$, $S$ and $T$.

- Let $R=\{1, 2, 3, 4\}$. Then $\displaystyle \sum\limits_{x\in R}x^2 = 1^2+2^2+3^2+4^2 = 30.$
- Let $S=\{1, 32, 1256, -23, 44\}$. Then $\displaystyle \sum\limits_{x \in S} x = 1+32+1256+(-23)+44=1310.$
- Let $T=\{a, b, c, d\}$. Then $\displaystyle \sum\limits_{t \in T} 5t = 5a+5b+5c+5d.$

This workbook produced by HELM is a good revision aid, containing key points for revision and many worked examples.

- Sigma notation at
**math**centre

$P[X=x]$ denotes the probability that the random variable $X$ takes on the value $x$. $P[X>x]$ denotes the probability that the random variable $X$ takes on a value greater than $x$. Similarly, $P[X

The binomial coefficient is defined as \[\binom nk = \frac{n!}{k! (n-k)!}\] where $n$ and $k$ are integers with $0\leq k \leq n$. It is so called as these numbers are the coefficients of the binomial expansion.

Another form for writing this is $^n \mathrm{C}_k$.

The notation $n!$ is explained on our page on factorials.