Home > Sequences and Series > Series Sigma Notation (basic)

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

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