1.1 Sigma notation, finite & Infinite series

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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 $\left\{{a}_{1};{a}_{2};{a}_{3};...;{a}_{n}\right\}$ :

$\begin{array}{c}\hfill {S}_{n}={a}_{1}+{a}_{2}+{a}_{3}+...+{a}_{n}\end{array}$

For example, if we have the following sequence of numbers

$\begin{array}{c}\hfill 1;4;9;25;36;49;...\end{array}$

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

$\begin{array}{c}\hfill {S}_{4}=1+4+9+25=39\end{array}$

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 :

$\begin{array}{c}\hfill {S}_{\infty }={a}_{1}+{a}_{2}+{a}_{3}+...\end{array}$

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:

$\sum _{i=m}^{n}{a}_{i}={a}_{m}+{a}_{m+1}+...+{a}_{n-1}+{a}_{n}$

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:

${S}_{n}=\sum _{i=1}^{n}{a}_{i}={a}_{1}+{a}_{2}+...+{a}_{n}$

For example, in the following sum,

$\sum _{i=1}^{5}i$

we have to add together all the terms in the sequence ${a}_{i}=i$ from $i=1$ up until $i=5$ :

$\sum _{i=1}^{5}i=1+2+3+4+5=15$

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. $\begin{array}{c}\hfill \sum _{i=3}^{10}\left(3{x}^{i}\right)=3{x}^{3}+3{x}^{4}+...+3{x}^{9}+3{x}^{10}\end{array}$
for any value $x$ .
Notice that in the second example we used three dots (...) to indicate that we had left out part of the sum. We do this to avoid writing out every term of a sum.

Some basic rules for sigma notation

1. Given two sequences, ${a}_{i}$ and ${b}_{i}$ ,
$\begin{array}{ccc}\hfill \sum _{i=1}^{n}\left({a}_{i}+{b}_{i}\right)& =& \sum _{i=1}^{n}{a}_{i}+\sum _{i=1}^{n}{b}_{i}\hfill \end{array}$
2. For any constant $c$ that is not dependent on the index $i$ ,
$\begin{array}{ccc}\hfill \sum _{i=1}^{n}c·{a}_{i}& =& c·{a}_{1}+c·{a}_{2}+c·{a}_{3}+...+c·{a}_{n}\hfill \\ & =& c\phantom{\rule{0.277778em}{0ex}}\left({a}_{1}+{a}_{2}+{a}_{3}+...+{a}_{n}\right)\hfill \\ & =& c\sum _{i=1}^{n}{a}_{i}\hfill \end{array}$

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 :

$\begin{array}{c}\hfill {a}_{n}={a}_{1}+d\left(n-1\right)\end{array}$

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:

$\begin{array}{ccc}\hfill {a}_{i}& =& {a}_{1}+d\left(i-1\right)\hfill \\ & =& 1+0·\left(i-1\right)\hfill \\ & =& 1\hfill \\ \hfill \left\{{a}_{i}\right\}& =& \left\{1;1;1;1;1;...\right\}\hfill \end{array}$

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

$\sum _{i=1}^{n}{a}_{i}=\sum _{i=1}^{n}1=1+1+1+...+1\phantom{\rule{2.em}{0ex}}\left(n\mathrm{times}\right)$

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 $n$ :

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