Introduction to Series and Sigma Notation

Source: Bedtime Stories of Maths

The infinite sums of a sequence of numbers (or terms) is called a series. A sequence forms a pattern, these sequences can be identified through the patterns.

One important series (which happen to occur naturally in plants and nature) is Fibonacci series. The series go like this:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...

As you can see, the number in any sequence is the sum of the two previous numbers - and it goes on forever!

A series follows a mathematical pattern and the terms can be described by a formula. We call any general or nth term of a series ${ T }_{ n }$. where n stands for the number of the term and must be a positive integer.

For example, for the series 3 + 8 + 13 + 18 + 23 + ... find ${ T }_{ n }$.

The first term is 3

Second term is 8 = 3 + 5 = 3 + 1 x 5

Third term is 13 = 3 + 5 + 5 = 3 + 2 x 5

Fourth term is 18 = 3 + 5 + 5 + 5 = 3 + 3 x 5

...and so on, following this pattern:

${ T }_{ n }=3+(n-1)\times 5\\ { T }_{ n }=3+5n-5\\ { T }_{ n }=5n-2$

Examples

1. The nth term of a series is given by $t_{ n }=5n-1$. Which term of the series is equal to 104?

$t_{ n }=5n-1\\ 5n-1=104\\ 5n=105\\ n=21$

2. The nth term of a series is given by ${u}_{n}=102-3n$. Find the value of n for the first negative term in the series.

For the first negative term, we want:

$u_{ n }\le 0\\ 102-3n\le 0\\ -3n\le -102\\ n\ge 34$

Since n is an integer, n=34 i.e. the 34th term is the first negative term.

3. The nth term of a series is given by the formula ${ t }_{ n }={ 2 }^{ n }-1$. Which term of the series is equal to 4095?

${ t }_{ n }={ 2 }^{ n }-1\\ \\ 4095={ 2 }^{ n }-1\\ \\ 4096={ 2 }^{ n }\\ \\ \log _{ 10 }{ 4096 } =\log _{ 10 }{ { 2 }^{ n } } \\ \\ \log _{ 10 }{ 4096 } =n\log _{ 10 }{ { 2 } } \\ \\ n=\cfrac { \log _{ 10 }{ 4096 } }{ \log _{ 10 }{ { 2 } } } \\ \\ n=12$

Sigma Notation

Series are often written in sigma notation ($\Sigma$) - which stands for the sum of a series. You may have seen this in finding the mean of a certain data in statistics.

$\bar { x } =\cfrac { \Sigma fx }{ \Sigma f }$

Examples

1. Evaluate $\sum _{ r=1 }^{ 5 }{ { r }^{ 2 } }$

$\sum _{ r=1 }^{ 5 }{ { r }^{ 2 } }$ means the sum of terms where the formula is ${r}^{2}$ with r starting at 1 and ending at 5.

So,

$\sum _{ r=1 }^{ 5 }{ { r }^{ 2 } } ={ 1 }^{ 2 }+{ 2 }^{ 2 }+{ 3 }^{ 2 }+{ 4 }^{ 2 }+{ 5 }^{ 2 }\\ \\ \qquad =1+4+9+16+25\quad =\quad 55$

2. Write 7 + 11 + 15 + ... + (4k + 3) in sigma notation.

$4n+3=4\times 1+3=7$
We can write the series as $\sum _{ 1 }^{ k }{ (4n+3) }$.
The number of terms in the series $\sum _{ p }^{ q }{ f(n) }$ is p - q + 1.