Calculus/Summation notation

Summation notation allows an expression that contains a sum to be expressed in a simple, compact manner. The uppercase Greek letter sigma, Σ, is used to denote the sum of a set of numbers.

Example

$\sum _{i=3}^{7}i^{2}=3^{2}+4^{2}+5^{2}+6^{2}+7^{2}$

Let $f$ be a function and $M,N$ are integers with $N<M$ . Then

$\sum _{i=N}^{M}f(i)=f(N)+f(N+1)+f(N+2)+\cdots +f(M)$ .

We say $N$ is the lower limit and $M$ is the upper limit of the sum.

We can replace the letter $i$ with any other variable. For this reason $i$ is referred to as a dummy variable. So...

$\sum _{i=1}^{4}i=\sum _{j=1}^{4}j=\sum _{\alpha =1}^{4}\alpha =1+2+3+4$

Conventionally we use the letters $i$ , $j$ , $k$ , $m$ for dummy variables.

Example

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

Here, the dummy variable is $i$ , the lower limit of summation is 1, and the upper limit is 5.

Example

Sometimes, you will see summation signs with no dummy variable specified, e.g.,

$\sum _{1}^{4}i^{3}=100$

In such cases the correct dummy variable should be clear from the context.

You may also see cases where the limits are unspecified. Here too, they must be deduced from the context.

Common summations

$\sum _{i=1}^{n}c=c+c+\cdots +c=nc\ ,\ c\in \mathbb {R}$

$\sum _{i=1}^{n}i=1+2+3+\cdots +n={\frac {n(n+1)}{2}}$

$\sum _{i=1}^{n}i^{2}=1^{2}+2^{2}+3^{2}+\cdots +n^{2}={\frac {n(n+1)(2n+1)}{6}}$

$\sum _{i=1}^{n}i^{3}=1^{3}+2^{3}+3^{3}+\cdots +n^{3}=\left(\sum _{i=1}^{n}i\right)^{2}=\left({\frac {n(n+1)}{2}}\right)^{2}$

$\sum _{i=1}^{\infty }a_{i}=\lim _{t\to \infty }\left[\sum _{i=1}^{t}a_{i}\right]$