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# 1.3 Series

## 1.3 Series (EMCDV)

It is often important and valuable to determine the sum of the terms of an arithmetic or geometric sequence. The sum of any sequence of numbers is called a series.

Finite series

We use the symbol $${S}_{n}$$ for the sum of the first $$n$$ terms of a sequence $$\left\{{T}_{1}; {T}_{2}; {T}_{3}; \ldots;{T}_{n}\right\}$$:

${S}_{n}={T}_{1}+{T}_{2}+{T}_{3}+\cdots + {T}_{n}$

If we sum only a finite number of terms, we get a finite series.

For example, consider the following sequence of numbers

$1; 4; 9; 16; 25; 36; 49; \ldots$

We can calculate the sum of the first four terms:

${S}_{4}=1+4+9+16=30$

This is an example of a finite series since we are only summing four terms.

Infinite series

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

${S}_{\infty }={T}_{1}+{T}_{2}+{T}_{3}+ \cdots$

### Sigma notation (EMCDW)

Sigma notation is a very useful and compact notation for writing the sum of a given number of terms of a sequence.

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

For example,

$\sum _{n=1}^{5}{2n} = 2 + 4 + 6 + 8 + 10 = 30$

In general,

$\sum _{i=m}^{n}{T}_{i}={T}_{m}+{T}_{m+1}+\cdots +{T}_{n-1}+{T}_{n}$

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;

• $${T}_{i}$$ is a term of a sequence;

• the number of terms in the series $$= \text{end index} - \text{start index} + \text{1}$$.

The index $$i$$ increases from $$m$$ to $$n$$ by steps of $$\text{1}$$.

Note that this is also sometimes written as:

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

When we write out all the terms in a sum, it is referred to as the expanded form.

If we are summing from $$i=1$$ (which implies summing from the first term in a sequence), then we can use either $${S}_{n}$$ or $$\sum$$ notation:

${S}_{n}=\sum _{i=1}^{n}{a}_{i}={a}_{1}+{a}_{2}+\cdots +{a}_{n} \quad (n \text{ terms})$

## Worked example 4: Sigma notation

Expand the sequence and find the value of the series:

$\sum _{n=1}^{6}{2}^{n}$

### Expand the formula and write down the first six terms of the sequence

\begin{align*} \sum _{n=1}^{6}{2}^{n} &= 2^{1} + 2^{2} + 2^{3} + 2^{4} + 2^{5} + 2^{6} \quad (\text{6} \text{ terms}) \\ &= 2 + 4 + 8 + 16 + 32 + 64 \end{align*}

This is a geometric sequence $$2; 4; 8; 16; 32; 64$$ with a constant ratio of $$\text{2}$$ between consecutive terms.

### Determine the sum of the first six terms of the sequence

\begin{align*} S _{6} &= 2 + 4 + 8 + 16 + 32 + 64 \\ &= 126 \end{align*}

## Worked example 5: Sigma notation

Find the value of the series:

$\sum _{n=3}^{7}{2an}$

### Expand the sequence and write down the five terms

\begin{align*} \sum _{n=3}^{7}{2an} &= 2a(3) + 2a(4) + 2a(5) + 2a(6) + 2a(7) \quad (5 \text{ terms}) \\ &= 6a + 8a + 10a +12a + 14a \end{align*}

### Determine the sum of the five terms of the sequence

\begin{align*} S _{5} &= 6a + 8a + 10a +12a + 14a \\ &= 50a \end{align*}
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## Worked example 6: Sigma notation

Write the following series in sigma notation:

$31 + 24 + 17 + 10 + 3$

### Consider the series and determine if it is an arithmetic or geometric series

First test for an arithmetic series: is there a common difference?

We let:

$\begin{array}{rll} T_{1} &= 31; &T_{4} = 10; \\ T_{2} &= 24; &T_{5} = 3; \\ T_{3} &= 17; & \end{array}$

We calculate:

\begin{align*} d &= T_{2} - T_{1} \\ &= 24 - 31 \\ &= -7 \\ d &= T_{3} - T_{2} \\ &= 17 - 24 \\ &= -7 \end{align*}

There is a common difference of $$-7$$, therefore this is an arithmetic series.

### Determine the general formula of the series

\begin{align*} T_{n} &= a + (n-1)d \\ &= 31 + (n-1)(-7) \\ &= 31 -7n + 7 \\ &= -7n + 38 \end{align*}

Be careful: brackets must be used when substituting $$d = -7$$ into the general term. Otherwise the equation would be $$T_{n} = 31 + (n-1) - 7$$, which would be incorrect.

### Determine the sum of the series and write in sigma notation

\begin{align*} 31 + 24 + 17 + 10 + 3 &= 85 \\ \therefore \sum _{n=1}^{5}{(-7n + 38)} &= 85 \end{align*}

#### Rules for sigma notation

1. Given two sequences, $${a}_{i}$$ and $${b}_{i}$$:

$\sum _{i=1}^{n}\left({a}_{i}+{b}_{i}\right) = \sum _{i=1}^{n}{a}_{i}+\sum _{i=1}^{n}{b}_{i}$
2. For any constant $$c$$ that is not dependent on the index $$i$$:

\begin{align*} \sum _{i=1}^{n} (c \cdot {a}_{i}) & = c\cdot{a}_{1}+c\cdot{a}_{2}+c\cdot{a}_{3}+\cdots +c\cdot{a}_{n} \\ & = c \left({a}_{1}+{a}_{2}+{a}_{3}+\cdots +{a}_{n}\right) \\ & = c\sum _{i=1}^{n}{a}_{i} \end{align*}
3. Be accurate with the use of brackets:

Example 1:

\begin{align*} \sum _{n=1}^{3}{(2n + 1)}& = 3 + 5 + 7 \\ & = 15 \end{align*}

Example 2:

\begin{align*} \sum _{n=1}^{3}{(2n) + 1}& = (2 + 4 + 6) + 1 \\ & = 13 \end{align*}

Note: the series in the second example has the general term $$T_{n} = 2n$$ and the $$\text{+1}$$ is added to the sum of the three terms. It is very important in sigma notation to use brackets correctly.

4. $\sum_{i = m}^{n}{a_{i}}$

The values of $$i$$:

• start at $$m$$ ($$m$$ is not always $$\text{1}$$);
• increase in steps of $$\text{1}$$;
• and end at $$n$$.
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## Sigma notation

Textbook Exercise 1.7

Determine the value of the following:

$\sum _{k=1}^{4}{2}$

$$2 + 2 + 2 + 2 = 8$$
$\sum _{i=-1}^{3}i$
\begin{align*} \sum _{i=-1}^{3}i & = -1 + 0 + 1 + 2 + 3 \\ & = 5 \end{align*}
$\sum _{n=2}^{5}(3n - 2)$
\begin{align*} \sum _{n=2}^{5}(3n - 2) & = [3(2) - 2] + [3(3) - 2] + [3(4) - 2] + [3(5) - 2] \\ & = 4 + 7 + 10 + 13 \\ &= 34 \end{align*}

Expand the series:

$\sum _{k=1}^{6}{0^{k}}$
$$0^{1} + 0^{2} + 0^{3} + 0^{4} + 0^{5} + 0^{6} = 0$$
$\sum_{n=-3}^{0}{8}$
$$8 + 8 + 8 + 8 = 32$$
$\sum _{k=1}^{5}(ak)$
\begin{align*} \sum _{k=1}^{5}(ak) & = a + 2a + 3a + 4a + 5a \\ & = 15a \end{align*}

Calculate the value of $$a$$:

$\sum _{k=1}^{3} \left( a \cdot {2}^{k-1} \right) =28$
\begin{align*} \sum _{k=1}^{3} \left( a \cdot {2}^{k-1} \right) &=28\\ \therefore a+ 2a + 4a & = 28 \\ 7a &= 28 \\ \therefore a &= 4 \end{align*}
$\sum _{j=1}^{4} \left( 2^{-j} \right) = a$
\begin{align*} \sum _{j=1}^{4} \left( 2^{-j} \right) &= a \\ \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} & = a \\ \therefore \frac{15}{16} &= a \end{align*}

Write the following in sigma notation:

$\frac{1}{9} + \frac{1}{3} + 1 + 3$

Geometric series with $$a = \frac{1}{9}$$, constant ratio $$r = 3$$ and general formula $$T_{n} = ar^{n-1}$$.

\begin{align*} T_{n} &= ar^{n-1} \\ & = \frac{1}{9}(3)^{n-1} \\ &= 3^{-2} \cdot 3^{n-1} \\ &= 3^{n-3} \\ \therefore & \sum _{n=1}^{4} \left( 3^{n-3} \right) \end{align*}

Write the sum of the first $$\text{25}$$ terms of the series below in sigma notation:

$11 + 4 - 3 - 10 \ldots$

Arithmetic series with $$a = 11$$, common difference $$d = -7$$ and general formula $$T_{n} = a + (n-1)d$$.

\begin{align*} T_{n} &= a + (n-1)d \\ & = 11 + (n-1)(-7)\\ &= 11 -7n + 7 \\ &= 18 - 7n \\ \therefore & \sum _{n = 1}^{25} \left( 18 - 7n \right) \end{align*}

Write the sum of the first $$\text{1 000}$$ natural, odd numbers in sigma notation.

$1 + 3 + 5 + 7 + \ldots$

Arithmetic series with $$a = 1$$, common difference $$d = 2$$ and general formula $$T_{n} = a + (n-1)d$$.

\begin{align*} T_{n} &= a + (n-1)d \\ & = 1 + (n-1)(2)\\ &= 1 + 2n - 2 \\ &= 2n - 1 \\ \therefore & \sum _{n=1}^{1000} \left( 2n - 1 \right) \\ \text{ or } & \sum _{n=0}^{999} \left( 2n + 1 \right) \end{align*}