sequences-01.md

% 6G4Z3001_1314 \\ Mathematics Fundamentals % Killian O'Brien % Oct 2013

Introduction

A quick trip through the unit

See the unit's Moodle area for info on:

• Syllabus
• Teaching team & pattern
• Assessment
• Resources

Some motivation

Motivation \\ Convergence and divergence in applied mathematics

• Convergence is the concept of an (infinite) process getting closer and closer to some limiting state.
• Divergence is where the process does not converge.

This is important to understand as it is fundamental to much of applied mathematics, e.g.

• Mathematical models of the real world are almost always only approximations and so only offer approximations to the truth.
• This is ok as long as we can in principle make the approximations as good as we want, i.e. make them converge to the actual truth.

Figure taken from A National Strategy for Advancing Climate Modeling by US National Academy of Sciences

In addition ...

• The mathematical problems set up on these already approximate models are almost always impossible to solve exactly ...
• ... and we can only find approximate solutions to them (so called numerical solutions).
• This is ok as long as we can in principle make the approximations as good as we want, i.e. make them converge to the actual solution.
• For time based models, such as weather/climate, these approximations get poorer and poorer as we look into the future, i.e. they diverge from the actual solution.

Motivation \\ Convergence and divergence in pure mathematics

• The integers $\mathbb{Z}$ and rationals $\mathbb{Q}$ can be rigorously described without too much trouble, $$ \mathbb{Z} = \left { \dots, -3, -2, -1, 0 , 1, 2, 3, \dots \right } \textit{ positive and negative whole numbers},$$ $$ \mathbb{Q} = \left { , \frac{n}{m} , : , n,m \in \mathbb{Z}, m \neq 0 \right } \textit{ ratios of integers} .$$

(See Piotr's Set Theory lectures for explanation of this ${ \quad }$notation.)

• Non-rational numbers (so called irrationals) are harder to describe rigorously and precisely, e.g. $$\sqrt{2} , , \pi , , e $$

• One way to do so is to describe them as the limits of infinite sequences or infinite sums of rational numbers, e.g. $$ \frac{\pi}{4} = \sum_{n=1}^\infty \frac{(-1)^{n+1}}{2n-1} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \dots $$ Many such interesting summation formulas exist for $\pi$.

• Infinite sequences and sums of mathematical objects and their convergence / divergence properties are a fundamental part of calculus and analysis: the study of the number line (2-d plane, 3-d space, ...) and functions defined on them.

Motivation \\ Convergence and divergence in pure & applied mathematics

• So there is lots of convergence & divergence going on
• This needs to be studied and understood
• We will study sequences and series and their convergence/divergence.

Sequences \\ Definitions and notation

Definition (1.1) A sequence is a list of elements (usually numbers) indexed by the positive integers. We usually use a single letter, with subscript, to denote the elements of a sequence, as in $$x_1 , x_2 , x_3 , x_4, \dots .$$ We can represent the sequence $x_1 , x_2 , x_3 , x_4, \dots$ using the compact notation $\left { x_n \right }_{n=1}^\infty$ or just simply $\left { x_n \right }$ if the $n$ indexing is understood.

Example (1.1)

• The sequence of positive integers $1,2,3, \dots $, or $\left { z_m \right }$, where the rule is $$z_m = m . $$

• The Fibonacci sequence: $1, 1, 2, 3, 5, 8, 13, ...$ or $\left {f_n \right }$ , where the rule is $$f_1 = 1, f_2 = 1 \text{ and } f_m = f_{m-1}+f_{m-2} \text{ for all } m \geq 2, $$ i.e. each element is the sum of the previous two elements. This type of rule for a sequence, where each term is defined using the values of some previous terms, is called a recurrence relation.

• The sequence of odd positive integers: $1, 3, 5, 7, ... $or $\left {a_n\right }$, where $$a_n = 2n-1.$$

• The sequence of prime numbers: $2, 3, 5, 7, 11, 13, ... $ or $\left { p_m \right }$ , where $$p_m = ???.$$

Sequences \\ Arithmetic and geometric sequences

Definition (1.2) An arithmetic sequence has a common difference between successive elements and a geometric sequence has a common ratio of successive elements. So we can write an arithmetic sequence as $$a, a + d , a + 2d , a + 3d ,\dots $$ or $\left { a_n\right }$ , where $a_n = a + (n - 1)d$ . Here $a$ is the initial element and $d$ is the common difference. A geometric sequence can be written as $$a, ar , ar^2 , ar^3 ,\dots $$ or $\left { a_n \right }$ , where $a_n = ar^{n-1}$. Here $a$ is the initial element and $r$ is the common ratio.

Examples (1.2 through 1.5)

• The sequences of positive integers and odd positive integers above are actually arithmetic sequences. The sequence of positive integers can be written as $\left { a_n \right }$ , where $$a_n = n = 1 + (n - 1),$$ so 1 is the initial element and 1 is the common difference. The sequence of odd positive integers can be written as $\left { a_n \right }$ , where $$a_n = 1 + (n - 1)2 ,$$ so 1 is the initial element and 2 is the common difference.

• The sequence, $$ 1, \frac{1}{2}, \frac{1}{4} , \frac{1}{8}, \dots $$ of powers of $\frac{1}{2}$, is geometric as it has the form $\left {a_n \right }$ , where $$a_n = \left ( \frac{1}{2} \right )^{n-1},$$ so 1 is the initial element and 2 the common ratio.

• Compound interest awarded on investments or charged on debts provide examples of geometric sequences. Suppose a bank offers an annual compound interest rate of $r %$ on an initial deposit of $P$ units. Then the value of the deposit after $n$ years is given by $$v_n = P \left ( 1 + \frac{r}{100} \right )^n .$$

• The sequence $\left { x_n \right }$, where $$x_n = \frac{1}{n^2} ,$$ is neither arithmetic nor geometric, as it is not possible to express it in the required form.

Sequences \\ Monotonic sequences: increasing or decreasing behaviour

Definition (1.3) A sequence $\left { x_n \right }$ is increasing if its elements satisfy $$ x_1 \leq x_2 \leq x_3 \leq \dots $$ and decreasing if they satisfy $$ x_1 \geq x_2 \geq x_3 \geq \dots .$$ If the inequalities are all strict then the sequence can be called strictly increasing or strictly decreasing as appropriate.

Proving monotonicity, i.e. increasing or decreasing behaviour

Two approaches ...

• Examine the difference between consecutive terms, i.e. $a_{n+1} - a_n$ and then try to establish one of the inequalities $$ a_{n+1} - a_n \geq 0 \quad \text{ or } \quad a_{n+1} - a_n \leq 0 .$$

• Examine the quotient of consecutive terms, $$ \frac{a_{n+1}}{a_n} ,$$ and try to establish one of the inequalties $$ \frac{a_{n+1}}{a_n} \geq 1 \quad \text{ or } \quad \frac{a_{n+1}}{a_n} \leq 1 .$$

• Strict inequalities here will establish the strict version of increasing / decreasing as appropriate.

Example (1.6) Investigate the sequence ${ a_n }$ defined by $$ a_n = \frac{5^n}{n!},$$ and describe its increasing / decreasing behaviour.

NB: $n!$ is the factorial of $n$, defined by, $$n! = n \times (n-1) \times (n-2) \times \dots \times 3 \times 2 \times 1 .$$

Ivestigate by computing some values...

<script type="text/x-sage"> list_plot([(5^n)/factorial(n) for n in range(5)]) </script>

and then prove it.

The above computation is performed by a Sage cell. In Matlab, use the plot command.