Sequences and Series

Contents

Sequence

Young Mary was observing the motion of a rubber ball as she dropped it on a floor. She jotted down the height attained by the ball in each successive bounce. A pattern was detected in the values. Mary wrote the numbers, in order, on a paper and showed it to her teacher. The teacher told her that we she had ‘discovered’ is called a ‘Sequence’ in basic Arithmetic.

Definition

The sequence (or progression) is a list of objects, usually numbers, that are ordered and are bounded by a rule. The things to remember include, a Rule that defines the relation between objects, the order in which the objects are mentioned and the fact that repetition is allowed.

Let us have a look at some examples (The respective Rule is bold).

• a, b, c, d, …., x, y, z is a sequence of all alphabets from a to z.
• 1, 3, 5, 7, 9 is a sequence of first five positive odd numbers.
• Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune is a sequence of planets in solar system with respect to distance from the Sun.

It should be remembered that the Rule can be anything that defines the nature’ of the sequence.

• Interestingly, the sequences D, R, A, W, E, R and R, E, W, A, R, D are two entirely different sequences. (Order matters!)

The sequence can be finite or infinite. For example,

• {10, 11, 12, 20, 21, 22, 30, 31, 32, 40, ….} is an infinite sequence, whereas,
• {2, 4, 8, 16} is a finite sequence.

Elements may repeat in the sequence e.g. {0, 3, 6, 0, 3, 6, 0, 3, 6}.

Mathematical Notation

Suppose we are given an infinite sequence {1, 1/2, 1/3, 1/4….}. This form is not something that you would like to deal with often. It is natural to think of a way to represent it in some ‘compact’ mathematical form that is easy to comprehend and work with. In different books on mathematics, you will find a sequence represented in the general form of its nth term (where n represents the index/number/position of the term in the sequence). In the given sequence, the $1^{st}$, $2^{nd}$ and $3^{rd}$ terms would be 1, 1/2 and 1/3 respectively.

It can be observed that for the given sequence, the numerator is a constant number i.e. 1, whereas the denominator varies with the value of index (n). In fact, denominator is 1 for n=1, 2 for n=2, 3 for n=3 and so on. This information is enough to construct a general term that can be written as:

$a_{n}=\frac{1}{n}$, n from $1\,to\,\infty$

Or,

$(a_{n})_{1 }^{\infty }=\frac{1}{n}$

Or,

$\left \{ a_{n} \right \}_{1 }^{\infty }=\frac{1}{n}$

Note that a sequence can be represented using braces {} or round brackets ().

The $a_{n}$ is also called the ‘nth term of the sequence’. By iterating over the value of n and plugging it in the nth term, the entire sequence can be constructed.

Finding the nth term of a sequence is of great significance and is introduced in many textbooks. As a practice exercise, the reader is encouraged to find the respective general terms of the following sequences:

• {12, 122, 1222, 12222….},
• {100, 99, 97, 94, 90},
• {1.0, -1.1, 1.2, -1.3, 1.4, -1.5…}

It is worth mentioning here that the general term gives us complete information about the sequence i.e. it essentially defines the rule for the sequence. On the other hand, representing a sequence as, for example, {3, 5, 7….} does not always provide the complete picture ({3, 5, 7…} can be defined as the sequence of odd numbers greater than 1 or as the sequence of prime numbers greater than 2.)

Some Specific Sequences

In mathematics, there are certain sequences that are often encountered. They have been described next:

a) Arithmetic sequence

When the consecutive elements in a sequence differ only by addition (or subtraction) of a constant number, technically known as the ‘common difference (d)’, the sequence is known as the Arithmetic sequence.

For instance, {1, 2, 3, 4, 5, 6, 7} is a finite Arithmetic sequence with a common difference of 1. The common difference can be either positive or negative. An arithmetic sequence $\left \{ a_{1}, a_{1}+d, a_{1}+2d, a_{1}+3d... \right \}$ can be represented in the general form as:

$a_{n} = a_{1} + (n-1)d$

where,

n is the index$a_{n}$ is the $n^{th}$ termd is the common difference

b) Geometric sequence

When the consecutive elements in the sequence differ only by multiplication (or division) by a constant non-zero number, technically known as the ‘common ratio (r)’, the sequence is known as the Geometric sequence.

For instance, {4, 12, 36, ….} is an infinite Geometric sequence with a common ratio of 3. The common ratio can be either a positive or a negative real number. It is worth noting that {2, -2, 2, -2, 2} is an example of a geometric sequence with a common ratio of -1. Geometric sequence $\left \{ a_{1}, a_{1}r, a_{1}r^{2}.... \right \}$ is represented as:

$a_{n} = a_{1}r^{n-1}$

where,

n is the index$a_{n}$is the$n^{th}$termr is the common ratio

Quick quiz: Is the sequence {1, 0, 0, 0, 0 ….}, a geometric sequence? If yes, what is the common ratio of this sequence?

Geometric sequence has many interesting applications; one of them is modelling the spread of a flu virus through a community (more about the problem at the end of this article).

c) Harmonic sequence

The sequence whose elements form an arithmetic sequence, when reciprocated. General term of a Harmonic sequence can be simply represented as:

$a_{n} = \frac{1}{a_{1}+(n-1)d}$

The sequence $\left \{ 1,-1,-\frac{1}{3}, -\frac{1}{5}, .... \right \}$ is a harmonic sequence with common difference, d, of -2.

d) Square sequence

It is the sequence {1, 4, 9, 16….}, where each element is square of a number (in ascending order). Its simplest representation is,

$a_{n} = n^{2}$

e) Fibonacci sequence

It is one of the most fascinating and elegant set of numbers. It is the sequence {0, 1, 1, 2, 3, 5, 8, 13….} with a surprisingly simple rule represented as:

$a_{n} = a_{n-2} + a_{n-1}$

where, 0 and 1 are 1st and 2nd elements of the sequence respectively.

The origin of the Fibonacci sequence dates back as early as 12th Century AD. Fibonacci sequence is argued to be observed abundantly in nature. For instance, the number of petals in flower species; 3 in mariposa lilies, 5 in grass of Parnassus, 8 in garden cosmos, 13 in some daisies, 21 in Chicory etc. In the famous Fibonacci rabbit problem, the population growth of rabbits can be predicted (under idealized, non-practical conditions).

Interesting fact: The ratio between the consecutive elements of the Fibonacci sequence converges to the famous Golden ratio equal to 1.61803398875. The reader is encouraged to construct a table with elements of Fibonacci sequence and compute the ratio of consecutive elements. The ratio will converge to the Golden ratio.

Arithmetic, Geometric and Harmonic mean

Now that we have the knowledge of Arithmetic, Geometric and Harmonic sequences, let us explore simple but notable concept. We shall then derive the relation between Arithmetic, Geometric and Harmonic means.

Definition: Given that x, M, y forms a sequence,

• M is said to be the Arithmetic Mean (AM) if x, M, y is an Arithmetic sequence.
• M is said to be the Geometric Mean (GM) if x, M, y is a Geometric sequence.
• M is said to be the Harmonic Mean (HM) if x, M, y is a Harmonic sequence.

Arithmetic Mean

Given an arithmetic sequence,

x, AM, y

Applying the concept of common difference,

AM – x = y – AM

$\frac{x+y}{2}= AM$

Geometric Mean

Given a Geometric sequence,

x, GM, y

Applying the concept of common ratio,

$\frac{GM}{x}=\frac{y}{GM}$

$xy = GM^{2}$

Harmonic Mean

Given a Harmonic sequence,

x, HM, y

As the reciprocals of the elements of a Harmonic sequence for an Arithmetic sequence,

$\frac{1}{x},\,\frac{1}{HM}, \,\frac{1}{y}$ (Arithmetic Sequence)

Now applying the concept of common difference,

$\frac{1}{HM} -\frac{1}{x}=\frac{1}{y}-\frac{1}{HM}$

$\frac{2}{HM} =\frac{x+y}{xy}$

Using the highlighted equations, the following equation can be derived:

$AM\times HM=GM^{2}$

These three represent the averages of their respective sequences. Each one of them has its own context of usage and should be properly implemented. AM is used when we are dealing with data that involves summation. GM is used where the data elements have some multiplicative relation e.g. in economics. HM is a bit deceptive. It is used when we are dealing with rates, such as average speed (rate of change of distance) of travel.

Series

…The teacher then asked Mary to find the total vertically descending distance the ball had covered. Mary gave it some thought and realized that it was simply the sum of the values she had noted. The teacher applauded and told Mary that in fact this process of adding the elements of the sequence is known as a ‘Series’.

Definition

The sum of the elements of a sequence is called its Series.

For instance, 1+2+3+4+5 …. is a series for the sequence {1, 2, 3, 4, 5….}. Like there are finite/infinite sequences, there are also finite/infinite series. The above given series is an example of an infinite series.

Series bring forth some exciting aspects. First, since series is a sum, therefore, the order of elements does not matter! (as opposed to a sequence). Second, the infinite series can be a Convergent or a Divergent series.

Convergent Series

If the sum of elements of infinite series ‘converges’ to a real number, the series is said to be a convergent series. One of the well-known convergent series is 1/2+1/4+1/8… which sums up to 1.

Divergent Series

If the sum of elements of infinite series does not converge to a real number, the series is said to be a divergent series. An example of divergent series is 2+4+8….

Quick Quiz: Is the series 1+1/2+1/3+1/4… convergent or divergent?

Series have profound applications in many areas of study in mathematics (both finite and infinite series), physics, finance, computer science etc. You will come across a number of series including the famous Taylor’s series, Binomial series etc.

Interesting fact: The Ramanujan Summation is the sum of all natural numbers starting from 1 to infinity. Surprisingly, the sum has been proved to converge to -1/12. The sum here, however, is not used in the traditional sense. There is great deal of knowledge present in the literature about this anomaly. This is only presented here to prompt some curiosity among the readers.

Sigma notation

The sum of elements (possibly infinite) of a sequence can be rather cumbersome to represent as a bunch of numbers with ‘+’ sign in between. It also makes mathematical manipulation difficult. What if there is some neat way of conveying the same information?

The sigma notation is there to make our lives easier. The sum of sequence, in other words, series, can be represented by simply putting the symbol $\Sigma$ with the general term of the sequence.

For instance, the series 1+2+3…. can be written as:

$\sum_{n=1}^{\infty }n$

The sigma notation is widely used in almost all areas of math. It is something you should be friend with.

Arithmetic Series

It is the sum of the arithmetic sequence. Given the general term of the arithmetic sequence:

$a_{k} = a_{1}+(k-1)d$

The arithmetic series can be represented as the sum of first n terms of {$a_{k}$}:

$S_{n} = \sum_{k=1}^{n}a_{k}=\sum_{k=1}^{n}(a_{1}+(k-1)d)$

Quick quiz: Use the above relation to sum first 100 natural numbers (and praise the power of maths).

Geometric Series

It is the sum of the elements of the geometric sequence. Given the general term of the geometric sequence:

$a_{k} = a_{1}r^{k-1}$

The geometric series can be represented as:

$S_{n} = \sum_{k=1}^{n}a_{k}=\sum_{k=1}^{n}a_{1}r^{k-1}$

Simplifying the expression (adjusting the limits) yields,

$S_{n} = a_{1}\sum_{k=0}^{n-1}r^{k}=a_{1}\frac{1-r^{n}}{1-r}$

If -1 < r < 1, the sum would converge as n approaches infinity, given by $\frac{1}{1-r}$.

Harmonics Series

It is the sum of the elements of the harmonic sequence. Given the general term of the harmonic sequence,

$a_{k} = \frac{1}{a_{1}+(k-1)d}$

The harmonic series can be represented as,

$S_{n} = \sum_{k=1}^{n}a_{k}=\sum_{k=1}^{n}\frac{1}{a_{1}+(k-1)d}$

With d=1, this series seems to be convergent at first, since each following term shrinks in size and approaches 0 as n approaches infinity. But after introducing tests for convergence, it can be shown that the series is divergent.

Solved Examples

As promised, we now go through an example of a disease outbreak. Suppose that a flu is spreading through a population. Initially, only one person is infected with the flu. It is also known that an infected person will infect two persons after one round of infection (and will not infect more people).  Therefore, after first round of infection, there are two newly infected person each of which will infect two persons further and this continues unbounded (simplified scenario). Our task is to derive the mathematical form of the rule that tells,

1. Newly infected persons after n rounds of infection
2. The total number of infected persons after n rounds of infection

1) The first step is to get the idea of how the sequence of newly infected persons look like. Initially, we have 1 person (0th round), then two newly infected person (total 3), then 4 newly infected person (total 7), and so on. Therefore, the sequence comes out to be:

{1, 2, 4, 8,16…}

It can be written in a compact form simply as:

$\left \{ a_{n} \right \}=2^{n}$, n starts from 0

This is the solution of part 1. Note that it is a geometric sequence with r=2

2) To find the total number of infected people after n rounds, we must find the sum of the sequence up to the $n^{th}$ term. In other words, we should find the $S_{n}$ term that we explored in the ‘Series’ section.

$S_{n} = \sum_{k=0}^{n}a_{k}=\sum_{k=0}^{n}2^{k}$

$=\frac{1-2^{n+1}}{1-2}$

This gives,

$S_{n} = 2^{n+1}-1$

This is the solution of part 2.

Note that using the answer derived in part 2, valuable information about the nature of the disease can be inferred. For instance, after 4 rounds of infection, a total of 31 people would have been infected, whereas, if we let the disease spread at the same rate, then in just next 5 rounds, a total of 1023 people would have been infected (more than 30 times!). It should also be noted that the value of r has great effects on the results. For instance, if r < 1, the disease will fade away. If r = 1, the number of patients will not increase, and the disease will not spread. If r > 1, the disease spreads. Greater the value of r, more rapidly the disease spreads.

Although this is an extremely simplified model of a disease spread. There are several factors, including people’s habits, limited population size, introduction of vaccine/medicine etc. that greatly impacts the results. In a more realistic approach, many of these factors are considered and included in the model.

Relative magnitudes of Arithmetic, Geometric and Harmonic means

Given p and q are distinct positive numbers, find the relation between relative magnitudes of their Arithmetic, Geometric and Harmonic means.

As derived earlier, the three means are represented as:

$AM = \frac{p+q}{2}$

$GM = \sqrt{pq}$

$HM = \frac{2pq}{p+q}$

It can be observed that:

AM > GM

And,

$HM = \frac{GM^{2}}{AM}=GM*(\frac{GM}{AM})=GM*a$

where, a < 1.

This gives,

HM < GM

Putting it together,

HM < GM < AM

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