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## Series Sigma Notation (basic)

A series can be described using sigma $\Sigma$ notation

So what does sigma notation look like? Let’s look at an example

$S=\displaystyle\sum_{r=3}^{7}r^2$

This tells us that we a dealing with series with terms that are of the form $r^2$

1. The first term is when $r=3$
2. The next is when $r=4$
3. We continue in this manner by adding one each time to r until r = 7
4. Each of these terms is added.

So we have $S=3^2+4^2+5^2+6^2+7^2=135$

### Summing a series given in sigma notation

The least technical method would be to use the one above.

Example

Find S, if $S=\displaystyle\sum_{r=5}^{8}(r+r^2)$

$\implies S=(5+5^2)+(6+6^2)+(7+7^2)+(8+8^2)$

$\implies S=30+42+56+72=200$

You have noticed that we could have written as a simplification:

$\implies S=(5+6+7+8)+(5^2+6^2+7^2+8^2)$

And it does follow that $\displaystyle\sum_{r=1}^{n}(f(r)+g(r))=\displaystyle\sum_{r=1}^{n}f(r)+\displaystyle\sum_{r=1}^{n}g(r)$

### Arithmetic representations in sigma notation

A sigma expression in the form $\displaystyle\sum_{r=1}^{n}(ar+b)$ where ‘a’ and ‘b’ are constants will produce an arithmetic series, with a common difference of ‘a’ and a starting value of ‘a+b’.  Using this imformation we could sum a series of this form using the formulae we learnt for arithmetic series.

Examples

Calculate $\displaystyle\sum_{r=1}^{10}(5r-2)$

This is an arithmetic series

$a=3$

$d=5$

$n=10$

$S_n=\frac{n}{2}(2a+(n-1)d)$

$\implies S_{10}=\frac{10}{2}(2\times 3+5\times 9)=255$

Calculate $\displaystyle\sum_{r=11}^{19}(-3r+10)$

This is an arithmetic series

$a=7$

$d=-3$

$S_n=\frac{n}{2}(2a+(n-1)d)$

The required value is $S_{19}-S_{10}=\frac{19}{2}(2\times 7-3\times 18)-\frac{10}{2}(2\times 7-3\times 9)=-315$

### Geometric representations in sigma notation

A sigma expression in the form $\displaystyle\sum_{r=1}^{n}(ab^r)$, where the first term is $a\times b$ and the common ratio is $b$.

Example

Calculate $\displaystyle\sum_{r=1}^{20}(4\times 2^r)$

This is a geometric series

$a=8$

$r=2$

$n=20$

$S_n=\frac{a(1-r^n)}{1-r}$

$S_{20}=\frac{8(1-2^{20})}{1-2}=8388600$