Summation

Summation is the addition of a sequence of numbers, expressed with sigma notation in the general form:

$$\sum _{k=n}^N{a}_k$$

where $n<N$. We read this as adding up the terms $a_k$ as $k$ ranges over the integers from $k=n$ as its lower index to $k=N$ as its upper index. Think of it as a for loop!

Closed formulas

We additionally have some closed formulas:

$$\sum _{k=1}^{n}k=\frac{n(n+1)}{2}$$ $$\sum _{k=1}^n{k}^2=\frac{n(n+1)(2n+1)}{6}$$ $$\sum _{k=1}^n{k}^3=\frac{n^2(n+1{)}^2}{4}$$

Properties of sigma notation

For a constant $c\in \mathbb{R}$, as a consequence of the distributive property of real numbers:

$$\sum _{k=n}^{N}c{a}_k=c\sum _{k=n}^N{a}_k$$

Due to the commutative property of real numbers, when we have addition or subtraction of terms:

$$\sum _{k=n}^N(a_k+b_k)=\sum _{k=n}^N{a}_k+\sum _{k=n}^N{b}_k$$