Summation Notation

We use sigma notation to represent the sums.

Properties of Summation (PS)

1. Multiplying by a constant

${\sum }_{i=m}^{n}c{x}_i=c{\sum }_{i=m}^n{x}_i$

2. Adding two sums and extracting two sums ${\sum }_{i=m}^n{x}_i+{\sum }_{i=m}^n{y}_i={\sum }_{i=m}^n(x_i+y_i)$ ${\sum }_{i=m}^n{x}_i-{\sum }_{i=m}^n{y}_i={\sum }_{i=m}^n(x_i-y_i)$

3. Changing the bounds of the index of summation (this one is a bit hard to wrap your head around)#to-review

${\sum }_{i=m}^n{x}_i={\sum }_{i=m+k}^{n+k}{x}_{i-k}$

Summation Formulas

Formulas that are useful for efficient evaluation. ${\sum }_{i=1}^{n}1=n$ ${\sum }_{i=1}^{n}c=cn$ ${\sum }_{i=1}^{n}i=\frac{n(n+1)}{2}$ ${\sum }_{i=1}^n{i}^2=\frac{n(n+1)(2n+1)}{6}$ ${\sum }_{i=1}^n{i}^3=(\frac{n(n+1)}{2}{)}^2$