1.1 Sigma notation, finite & Infinite series  (Page 6/4)

Series

In this section we simply work on the concept of adding up the numbers belonging to arithmetic and geometric sequences. We call the sum of any sequence of numbers a series .

Some basics

If we add up the terms of a sequence, we obtain what is called a series . If we only sum a finite amount of terms, we get a finite series . We use the symbol $S_n$ to mean the sum of the first $n$ terms of a sequence $\{a_1;a_2;a_3;...;a_n\}$ :

For example, if we have the following sequence of numbers

and we wish to find the sum of the first 4 terms, then we write

The above is an example of a finite series since we are only summing 4 terms.

If we sum infinitely many terms of a sequence, we get an infinite series :

Sigma notation

In this section we introduce a notation that will make our lives a little easier.

A sum may be written out using the summation symbol $\sum$ . This symbol is sigma , which is the capital letter “S” in the Greek alphabet. It indicates that you must sum the expression to the right of it:

where

• $i$ is the index of the sum;
• $m$ is the lower bound (or start index), shown below the summation symbol;
• $n$ is the upper bound (or end index), shown above the summation symbol;
• $a_i$ are the terms of a sequence.

The index $i$ is increased from $m$ to $n$ in steps of 1.

If we are summing from $n=1$ (which implies summing from the first term in a sequence), then we can use either $S_n$ - or $\sum$ -notation since they mean the same thing:

For example, in the following sum,

we have to add together all the terms in the sequence $a_i=i$ from $i=1$ up until $i=5$ :

Examples

1. $$\begin{array}{ccc}\hfill \sum _{i=1}^6{2}^i& =& 2^1+2^2+2^3+2^4+2^5+2^6\hfill \\ & =& 2+4+8+16+32+64\hfill \\ & =& 126\hfill \end{array}$$
2. $$\begin{array}{c}\hfill \sum _{i=3}^{10}\left(3x^i\right)=3x^3+3x^4+...+3x^9+3x^{10}\end{array}$$
   for any value $x$ .

Some basic rules for sigma notation

1. Given two sequences, $a_i$ and $b_i$ ,
   $$\begin{array}{ccc}\hfill \sum _{i=1}^n(a_i+b_i)& =& \sum _{i=1}^n{a}_i+\sum _{i=1}^n{b}_i\hfill \end{array}$$
2. For any constant $c$ that is not dependent on the index $i$ ,
   $$\begin{array}{ccc}\hfill \sum _{i=1}^{n}c·a_i& =& c·a_1+c·a_2+c·a_3+...+c·a_n\hfill \\ & =& c(a_1+a_2+a_3+...+a_n)\hfill \\ & =& c\sum _{i=1}^n{a}_i\hfill \end{array}$$

Exercises

1. What is ${\sum }_{k=1}^{4}2$ ?
2. Determine ${\sum }_{i=-1}^{3}i$ .
3. Expand ${\sum }_{k=0}^{5}i$ .
4. Calculate the value of $a$ if:
   $$\sum _{k=1}^{3}a·2^{k-1}=28$$

Finite arithmetic series

Remember that an arithmetic sequence is a set of numbers, such that the difference between any term and the previous term is a constant number, $d$ , called the constant difference :

where

• $n$ is the index of the sequence;
• $a_n$ is the $n^{th}$ -term of the sequence;
• $a_1$ is the first term;
• $d$ is the common difference.

When we sum a finite number of terms in an arithmetic sequence, we get a finite arithmetic series .

The simplest arithmetic sequence is when $a_1=1$ and $d=0$ in the general form [link] ; in other words all the terms in the sequence are 1:

If we wish to sum this sequence from $i=1$ to any positive integer $n$ , we would write

Since all the terms are equal to 1, it means that if we sum to $n$ we will be adding $n$ -number of 1's together, which is simply equal to $n$ :