Lecture 8

2.4 Sequences and Summations

Sequences

A sequence is a function from a subset of the set of natural numbers N or non-negative integers Z*

to a set S. We use the notation sn to denote the image of the integer n. We call sn a term of the

sequence.

N 1, 2, 3, 4, 5, ...

S s1, s2, s3, s4, s5, ...

Z* 0, 1, 2, 3, 4, ...

S s0, s1, s2, s3, s4, ...

It is important to be able to express the sequence S in closed form as a function. For example,

S = { 1, 4, 9, 16, 25, ...} has a closed form sn = n2.

S = { 1, 1/2, 1/3, 1/4, ...} is sn = 1/n.

S = { 0, 4, 18, 48, 100, ...} is sn = ?

Special Integer Sequences

1, 1/2, 1/4, 1/8, . . .

1, 3, 5, 7, . . .

1, -1, 1, -1, . . .

0, 4, 18, 48, 100, . . .

2, 5, 10, 17, 26, 37, . . .

- - - -

6

28

80

1800 4 18 48 100

4 14 30 52

10 16 22

6 6

Example: Method of Finite Differencescandidate for a Computer Program

What is the next number in the sequence?

2 5 10 17 26 37 3 5 7 9 11 2 2 2 2

1 2 3 4 5 6N

S

10nC5nC2nCS 321n )()()(

102

2n1n5

1

3n1n2

2

3n2n

))(())(())((

)()()( 2n3n53n4n56n5n 222

1nS 2n

An Algebraic Solutioncandidate for a Computer Program

Arithmetic Progression

An arithmetic progression is a sequence of the form

a, a+d, a+2d, . . . , a+nd

where the initial term a and the common difference d are real numbers. An arithmetic progression is a discrete analogue of the linear function

f(x) = dx + a.

a a+d a+2d a+3d . . . a + nd

d d d . . .

the difference between any successive pair of terms is equal to d

Geometric Progression

A geometric progression is a sequence of the form

a, ar, ar2, ... , arn

where the initial term a and the common ratio r are real numbers. A geometric progression is a discrete analogue of the exponential function

f(x) = arx.

a ar ar2 . . . ar(n-1) arn

r r . . . r

ar ar2 . . . arn

a ar ar(n-1)

the ratio of any successive pair of terms is equal to r

2n ns16941

n

1i

2i301451

Building a Pyramid

2513

51 ...

n

1i

2n

1i

22

n

1i

2

otherwiseii

3nfornS n

'

Summations

Summation notation below uses an integer variable i called the index of summation that runs through all integers starting with its lower limit m and ending with its upper limit n. The value of the term ai is computed for each i and added to the accumulating sum of all the terms

n

miia

Is is important to be able to express summations in closed forms as functions...

Geometric Series

The sum of terms of a geometric progression is called a geometric series. The closed form for a geometric progression (when a and r are real and r is not zero),

n

0i

1n

i

1rifa1n

1rif1r

aarar

)(

These are two important special cases of the geometric series...

finite series

infinite series

Binomial Identities

Pascal's Triangle

Recurrence Relationfor Combinations

n

0

1

2

3

4

5

m

0

1

2

3

4

m

1n

1m

1n

m

n

)!(!

!,

mnm

nmnC

n

n

Power Series

Exponential Trigonometric

These series are used in calculators and computer CPU's to compute exponential and

trigonometric functions. While the proofs of these formulae are the primary concern for

mathematics, their computational complexity and the possibility of improving their efficiency

are the primary concern in computer science...

Cardinality

The sets A and B have the same cardinality if and only if there is a one-to-one correspondence from A to B.

A set that is either finite or has the same cardinality as the set of positive integers is called countable. A set that is not countable is called uncountable.

Note: It is not a requirement that the one-to-one correspondence be known or provable, just that it exists. For more info on the continuing controversy regarding unprovable theorems in discrete mathematics, search on Cantor's Diagonalization Theorem, large cardinal axiom, Borel Diagonalization Theorem, and Axiom of Choice. For example see,

http://www.algebra.com/algebra/about/history/Talk:Axiom-of-choice.wikipedia

1 2 3 4 5 6 7 8 9 10 11 12 ...

1 3 5 7 9 11 13 15 17 19 21 23 ...

...

...

...

...

4

4

4

3

4

2

4

1

3

4

3

3

3

2

3

1

2

4

2

3

2

2

2

1

1

4

1

3

1

2

1

1

Rational Numbers are Countable

There is a one-to-one correspondence between the natural numbers and the rational numbers (assuming that n/m and qn/qm are considered to be different rational numbers.

Cantor's Diagonalization "Proof"

Cantor's proof requires the axiom of choice in which we assume that we can choose a value from a list in order to prove that the list does not exist.

We wish to prove that the real numbers between 0.0 and 1.0 are not countable. We first assume that there is a one-to-one correspondence between the natural numbers and the reals in the interval (0.0,1.0) as shown below.

1 0.d11 d12 d13 d14 ...

2 0.d21 d22 d23 d24 ...

3 0.d31 d32 d33 d34 ...

4 0.d41 d42 d43 d44 ...

: :

N reals in (0.0, 1.0)

where dij is the jth digit of the real number corresponding to the ith natural number. From this list we

construct a number,0.c1 c2 c3 c4 ...

where, ci = (dii +1) mod 10. This number clearly lies in the interval (0.0, 1.0) but is not a member

of the list above since its ith digit is different from the ith digit of the ith number for every number in the list.

Problems for Cantor's Proof

First of all this proof is not just a proof by contradiction. It has the additional requirement that in order to prove the contraction there must exist an entity which cannot exist if the proof is valid.

This is fundamentally different from demonstrating that if a statement S is true then a contradiction exists, which means that S must be false. In the case of Cantor's Proof we assume that the set of reals can be arranged into a list that contains all the members of the list. Then we assume that a value can be composed from the members of the list creating a new member in the set that is not in the list. Only if this impossible value exists can we prove that the list does not exist and, by extension the new member also cannot exist. This is one step removed from proof-by-contradiction and therefore requires an additional assumption (i.e. another axiom called the Axiom of Choice).*

0.4309411290814208725430924309842109...

0.7916109102982160219176902431932789...

0.2191190110012291029671642048275386...

0.1239010101902748657329873280783217...

0.4999999999999999999999999999999999...

In this example the composite number is 0.499999... But we must assume that there are infinitely many numbers in the list that do not contain any 9's therefore 0.4 followed by all 9's could not be the composite number. Since there is nothing special about the use of 9's in this example, we can make a similar argument about any particular composite number (refuting the axiom of choice), and therefore we cannot generate a value not in the list... Which means that we cannot prove anything about the countability of the set of reals in the interval (0.0, 1.0) using this method.

there is no well-orderingof the reals in this set...

*http://www.recipeland.com/encyclopaedia/index.php/Axiom_of_choice

Double Summations

4

1i

3

1jij3S

with ada.integer_text_io;

use ada.integer_text_io;

procedure double_sum_demo is

S : integer := 0;

begin

for i in 1..4 loop

for j in 1..3 loop

S := S + 3*i*j;

end loop;

end loop;

put(S,0);

end double_sum_demo;

180

72543618

i18

i9i6i3

ij3S

4

1i

4

1i

4

1i

3

1j

Some Useful Summation Formulae

A Numeric Example

Queuing TheoryCandidate for a Computer Program

Customers arrive randomly but at an average rate of customers per unit time. A server provides service at at rate of customers per unit time. If then this system will reach a steady state.

We will define the state of this system by the number of customers in the system. An empty system (no customers) is in state S0. When there is one customer the system is in state S1 and so on. In the steady state the rate at which customers enter the system is equal to the rate at which they leave the system.

Obtaining a Closed-Form Solution

by repeated substitution of Eq n into Eq n-1

since the system must be in some state the sum of all probabilities must equal 1

is strictly less than 1 so the closed form for the infinite series applies

the average number of customers in the system

again we apply the closed form

the average time a customer spends in the system