16. Sequences and Summations - Kangwoncs.kangwon.ac.kr/~ysmoon/courses/2011_1/dm/16.pdf · 2016-06-02 · Introduction 3.2 Sequences and Summations A sequence or series is just like

이산수학이산수학(Discrete Mathematics) (Discrete Mathematics) 수열과수열과 합합

(S d S ti )(S d S ti )(Sequences and Summations)(Sequences and Summations)

20112011년년 봄학기봄학기

강원대학교강원대학교 컴퓨터과학전공컴퓨터과학전공 문양세문양세강원대학교강원대학교 컴퓨터과학전공컴퓨터과학전공 문양세문양세

IntroductionIntroduction3.2 Sequences and Summations

A sequence or series is just like an ordered n-tuple (a1, a2, …, an), except:, n), p

• Each element in the sequences has an associated index number.(각 element는 색인(index) 번호와 결합되는 특성을 가진다.)

• A sequence or series may be infinite. (무한할 수 있다.)

• Example: 1, 1/2, 1/3, 1/4, …

A summation is a compact notation for the sum of all A summation is a compact notation for the sum of all terms in a (possibly infinite) series. ()

Discrete Mathematicsby Yang-Sae MoonPage 2

SequencesSequences3.2 Sequences and Summations

Formally: A sequence {an} is identified with a generating function f:SA for some subset SN (S=N or S=N{0}) and f f ( { })for some set A. (수열 {an}은 자연수 집합으로부터 A로의 함수…)

If f is a generating function for a sequence {a } then for If f is a generating function for a sequence {an}, then for nS, the symbol an denotes f(n).

Th i d f i (O ft i i d )The index of an is n. (Or, often i is used.)

S Af

123

a1 = f(1)a2 = f(2)a = f(3)3

4

a3 = f(3)a4 = f(4)


Sequence ExamplesSequence Examples3.2 Sequences and Summations

Example of an infinite series (무한 수열)

• Consider the series {a } = a1 a2 where (n1) a = f(n) = 1/n• Consider the series {an} = a1, a2, …, where (n1) an= f(n) = 1/n.

• Then, {an} = 1, 1/2, 1/3, 1/4, …

Example with repetitions (반복 수열)p p

• Consider the sequence {bn} = b0, b1, … (note 0 is an index) where bn = (1)n.

• {bn} = 1, 1, 1, 1, …

• Note repetitions! {bn} denotes an infinite sequence of 1’s and 1’s, p { n} q ,not the 2-element set {1, 1}.


Recognizing Sequences (1/2)Recognizing Sequences (1/2)3.2 Sequences and Summations

Sometimes, you’re given the first few terms of a sequence,

and you are asked to find the sequence’s generating

function, or a procedure to enumerate the sequence., p q(순열의 몇몇 값들에 기반하여 f(n)을 발견하는 문제에 자주 직면하게 된다.)

Examples: What’s the next number and f(n)?p f( )

• 1, 2, 3, 4, … (the next number is 5. f(n) = n

• 1, 3, 5, 7, … (the next number is 9. f(n) = 2n − 1


Recognizing Sequences (2/2)Recognizing Sequences (2/2)3.2 Sequences and Summations

Trouble with recognition (of generating functions)

• The problem of finding “the” generating function given just an

initial subsequence is not well defined. (잘 정의된 방법이 없음)

• This is because there are infinitely many computable functions that

will generate any given initial subsequencewill generate any given initial subsequence.(세상에는 시퀀스를 생성하는 셀 수 없이 많은 함수가 존재한다.)


What are Strings? (1/2)What are Strings? (1/2) -- skipskip3.2 Sequences and Summations

Strings are often restricted to sequences composed of symbols drawn from a finite alphabet, and may be indexed y p , yfrom 0 or 1. (스트링은 유한한 알파벳으로 구성된 심볼의 시퀀스이고, 0(or 1)부터 색인될 수 있다.)

More formally,

• Let be a finite set of symbols, i.e. an alphabet.y p

• A string s over alphabet is any sequence {si} of symbols, si, indexed by N or N{0}.

• If a, b, c, … are symbols, the string s = a, b, c, … can also be written abc …(i.e., without commas).

• If s is a finite string and t is a string, the concatenation of s with t, written st, is the string consisting of the symbols in s followed by the symbols in t


the symbols in t.

What are Strings? (2/2)What are Strings? (2/2) -- skipskip3.2 Sequences and Summations

More string notation

Th l gth | | f fi it t i g i it b f iti (i it • The length |s| of a finite string s is its number of positions (i.e., its number of index values i).

If s is a finite string and n N sn denotes the concatenation of n• If s is a finite string and nN, sn denotes the concatenation of ncopies of s. (스트링 s를 n번 concatenation하는 표현)

denotes the empty string the string of length 0• denotes the empty string, the string of length 0.

• If is an alphabet and nN,

− n {s | s is a string over of length n} (길이 n인 스트링)

− * {s | s is a finite string over } (상에서 구현 가능한 유한 스트링)


Summation NotationSummation Notation3.2 Sequences and Summations

Given a sequence {an}, an integer lower bound j0, and an

integer upper bound kj then the summation of {a } from integer upper bound kj, then the summation of {an} from

j to k is written and defined as follows:({a }의 j번째에서 k번째까지의 합 즉 a 로부터 a 까지의 합)({an}의 j번째에서 k번째까지의 합, 즉, aj로부터 ak까지의 합)

kjjk

i

k

i a...aa:aa 1

H i i ll d th i d f ti

kjjji iji

i a...aa:aa

1

Here, i is called the index of summation.


Generalized SummationsGeneralized Summations3.2 Sequences and Summations

For an infinite series, we may write:

...aa:a jjji

i 1

To sum a function over all members of a set X={x1, x2, …}:(집합 X의 모든 원소 x에 대해서)

...)x(f)x(f:)x(fXx

21

Or, if X={x|P(x)}, we may just write:(P( )를 t 로 하는 모든 에 대해서)(P(x)를 true로 하는 모든 x에 대해서)

...)x(f)x(f:)x(f)(P

21


)x(P

Summation ExamplesSummation Examples3.2 Sequences and Summations

A simple example

4

2 2 2 2

21 2 1 3 1 4 1

ii

( ) ( ) ( )2

4 1 9 1 16 15 10 1732

i

( ) ( ) ( )

An infinite sequence with a finite sum:

32

An infinite sequence with a finite sum:

21222 41

2110

0

......i

i

Using a predicate to define a set of elements to sum over:

874925947532x 22222 Discrete Mathematicsby Yang-Sae MoonPage 11

874925947532x10) prime is (

xx

Summation Manipulations (1/2)Summation Manipulations (1/2)3.2 Sequences and Summations

Some useful identities for summations:

( ) ( )cf x c f x (Distributive law)

( ) ( )x xcf x c f x

( ) ( ) ( ) ( )x x xf x g x f x g x (Application of

commutativity)

( ) ( )k k n

f i f i n

( ) ( )i j i j nf i f i n

(Index shifting)


Summation Manipulations (2/2)Summation Manipulations (2/2)3.2 Sequences and Summations

Some more useful identities for summations:

1( ) ( ) ( ) if

k m k

i j i j i mf i f i f i j m k

(Series splitting)1i j i j i m

0( ) ( )

k jk

i j if i f k i

(Order reversal)

2k k

2

0 0( ) (2 ) (2 1)

k k

i if i f i f i

(Grouping)


An Interesting ExampleAn Interesting Example3.2 Sequences and Summations

“I’m so smart; give me any 2-digit number n, and I’ll add

ll h b f 1 i h d i j f all the numbers from 1 to n in my head in just a few

seconds.” (1에서 n까지의 합을 수초 내에 계산하겠다!)n

I.e., Evaluate the summation:

n

i

i1

There is a simple formula for the result, discovered by p , y

Euler at age 12!


Euler’s Trick, IllustratedEuler’s Trick, Illustrated3.2 Sequences and Summations

Consider the sum:

1 + 2 + … + (n/2) + ((n/2)+1) + … + (n-1) + n

n+1…

n+1n+1n+1

nn/2 pairs of elements, each pair summing to /2 pairs of elements, each pair summing to nn+1, for a +1, for a

total of (total of (nn/2)(n+1). /2)(n+1). (합이 n+1인 두 쌍의 element가 n/2개 있다.)

21)n(n

in


21i

Symbolic Derivation of Trick (1/2) Symbolic Derivation of Trick (1/2) -- skipskip3.2 Sequences and Summations

))k(i(iiiii)k(n

i

k

i

n

ki

k

i

k

i

n

i

11

0111

2

11

))k()i))k(n((i)k(nk

iikiiii

111

011111

kk

jk

i

k

ji

)ik(f)i(f

)ik(f)i(f0

))i((i)i(i

))()))(((

knk)k(nk

ii

11

01

ii

)ik(f)i(f00

))i(n(i)in(i

kkknk

iiii

11101

...)in(i)in(ik

i

k

i

kn

i

k

i

111111


kn 2 since

Symbolic Derivation of Trick (2/2) Symbolic Derivation of Trick (2/2) -- skipskip3.2 Sequences and Summations

kkkn 1 1

1111

)ini()in(iik

i

k

i

k

i

n

i

111 21

)n()n(k)n( nk

i

211

/)n(ni

So, you only have to do 1 easy multiplication in your head,

then cut in half.

Also works for odd n (prove it by yourself).


Geometric Progression (Geometric Progression (등비수열등비수열))3.2 Sequences and Summations

A geometric progression is a series of the form a, ar, ar2, 3 k h Rar3, …, ark, where a,rR.

The sum of such a sequence is given by:q g y

k

iarS

We can reduce this to closed form via clever manipulation

i 0

We can reduce this to closed form via clever manipulation

of summations...

1 11

kki rS ar a


0 1i r

Derivation of Geometric Sum (1/3) Derivation of Geometric Sum (1/3) -- skipskip3.2 Sequences and Summations

Sn

iarS

nnnn

i

i

0

rararrrararrrS

nnn

i

i

i

i

i

i

i

i

11

0

1

000

arararn

i

in

i

)i(n

i

i

1

1

1

1

11

0

1

...arararar nn

i

in

ni

in

i

i

1

1

1

11


Derivation of Geometric Sum (2/3) Derivation of Geometric Sum (2/3) -- skipskip3.2 Sequences and Summations

ni

ni 00

arar)arar(ararrS n

i

in

i

i 1

1

001

1

arararar nn

i

i 01

1

0

aararar nn

i

i

i

i 1

1

0

0

)r(aS)r(aar nnn

i

ii

11 11

10

)()(i 0


Derivation of Geometric Sum (3/3)Derivation of Geometric Sum (3/3) -- skipskip3.2 Sequences and Summations

)r(aSrS n 11

)r(aSrS

)r(aSrSn 1

11

r

)r(a)r(Sn

n

1

111

1

rr

raS 1 en wh

11

a)n(aaarSrn

i

n

i

in

i

i 111 ,1 when000


Nested SummationsNested Summations3.2 Sequences and Summations

These have the meaning you’d expect.

32144 34 34 3

ijiijij

432166644

11 11 11 1

)(ii

ii ji ji j

6010611

ii


Some Shortcut ExpressionsSome Shortcut Expressions3.2 Sequences and Summations

Sum Closed Form

1 rarn

k )r(a n 11

n

k

10

r,ark )r( 1

1)n(n k

k1

n

k2

2)(

121 )n)(n(n

n

k3

k

k1 6

))((

1 22 )n(n k

k1

1

xxk

4)(

110

x,xk

11

xkxk

x1

1Infinite series

(무한급수)


11

x,kxk

21 )x(

Infinite Series (Infinite Series (무한급수무한급수) (1/2) ) (1/2) -- skipskip3.2 Sequences and Summations

10

x,x

nn

11k

• Let a = 1 and r = x, then 11

0

1

k

n

kn

xx

kx

n

kj aar 1

i

jj

rar

0 1 since

• If k , then xk+1 0

• Therefore, xn

1

1Therefore,

xxx

n 1

10


Infinite Series (Infinite Series (무한급수무한급수) (2/2) ) (2/2) -- skipskip3.2 Sequences and Summations

10

1

x,kx

nn

xn

n

11

0 xn 10

dx

d n 1 d

1

xdxx

dx nn

10

1

recall)x('g)x(f)x(g)x('f

)(fd

nxxdxd nn

211

11

)x(nx

nn

2))x(g()(g)(f)(g)(f

)x(fdx


Using the ShortcutsUsing the Shortcuts3.2 Sequences and Summations

Example: Evaluate .

100

50

2

k

k

• Use series splitting.

50k

1002

492

1002

• Solve for desired

summation.

kkkkkk

49100100

50

2

1

2

1

2

• Apply quadratic

series rule

kkkkkk

49

1

2100

1

2100

50

2

series rule.

• Evaluate. 6995049

6201101100

.,,,

92529742540350338


Cardinality: Formal DefinitionCardinality: Formal Definition3.2 Sequences and Summations

For any two (possibly infinite) sets A and B, we say that A

and B have the same cardinality (written |A|=|B|) iff and B have the same cardinality (written |A|=|B|) iff

there exists a bijection (bijective function) from A to B.(집합 A에서 집합 B로의 전단사함수가 존재하면 A와 B의 크기는 동일하다 )(집합 A에서 집합 B로의 전단사함수가 존재하면, A와 B의 크기는 동일하다.)

When A and B are finite, it is easy to see that such a

function exists iff A and B have the same number of function exists iff A and B have the same number of

elements nN.(집합 A B가 유한집합이고 동일한 개수의 원소를 가지면 A와 B가 동일한 크기(집합 A, B가 유한집합이고 동일한 개수의 원소를 가지면, A와 B가 동일한 크기

임을 보이는 것은 간단하다.)


Countable versus UncountableCountable versus Uncountable3.2 Sequences and Summations

For any set S, if S is finite or if |S|=|N|, we say S is

countable Else S is uncountablecountable. Else, S is uncountable.(유한집합이거나, 자연수 집합과 크기가 동일하면 countable하며, 그렇지 않으

면 uncountable하다.)면 하다 )

Intuition behind “countable:” we can enumerate

( ti ll li t) l t f S E l N Z(sequentially list) elements of S. Examples: N, Z.(집합 S의 원소에 번호를 매길 수(순차적으로 나열할 수) 있다.)

Uncountable means: No series of elements of S (even an

infinite series) can include all of S’s elements.

Examples: R, R2

(어떠한 나열 방법도 집합 S의 모든 원소를 포함할 수 없다. 즉, 집합 S의 원소에


번호를 매길 수 있는 방법이 없다.)

Countable Sets: Examples Countable Sets: Examples 3.2 Sequences and Summations

Theorem: The set Z is countable.

P f C id f Z N h f(i) 2i f i 0 d f(i) 2i 1 f i 0 • Proof: Consider f:ZN where f(i)=2i for i0 and f(i) = 2i1 for i<0.

Note f is bijective. (…, f(2)=3, f(1)=1, f(0)=0, f(1)=2, f(2)=4, …)

Theorem: The set of all ordered pairs of natural numbers

(n,m) is countable.

(1,1)

(1 2)

(2,1)

(2 2)

(3,1)

(3 2)

(4,1)

(4 2)

(5,1)

(5 2)

… consider sum is 2, thenconsider sum is 3, thenconsider sum is 4, then

(1,2)

(1,3)

(2,2)

(2,3)

(3,2)

(3,3)

(4,2)

(4,3)

(5,2)

(5,3)

…

…

consider sum is 5, thenconsider sum is 6, thenconsider …

(1,4)

(1,5)

(2,4)

(2,5)

(3,4)

(3,5)

(4,4)

(4,5)

(5,4)

(5,5)

…

…Note a set of rational numbers is countable!


… … … … … …

Uncountable Sets: Example (1/2) Uncountable Sets: Example (1/2) –– skipskip3.2 Sequences and Summations

Theorem: The open interval

[0,1) : {rR| 0 r < 1} is uncountable. ([0,1)의 실수는 uncountable)[0,1) : {rR| 0 r < 1} is uncountable. ([0,1)의 실수는 uncountable)

Proof by Cantor

• Assume there is a series {ri} = r1, r2, ... containing all elements r[0,1).

• Consider listing the elements of {ri} in decimal notation in order of

increasing index:

r1 = 0.d1,1 d1,2 d1,3 d1,4 d1,5 d1,6 d1,7 d1,8…

r2 = 0.d2,1 d2,2 d2,3 d2,4 d2,5 d2,6 d2,7 d2,8…

r3 = 0.d3,1 d3,2 d3,3 d3,4 d3,5 d3,6 d3,7 d3,8…

0 d d d d d d d dr4 = 0.d4,1 d4,2 d4,3 d4,4 d4,5 d4,6 d4,7 d4,8…

…


• Now, consider r’ = 0.d1 d2 d3 d4 … where di = 4 if dii 4 and di = 5 if dii = 4.

Uncountable Sets: Example (2/2)Uncountable Sets: Example (2/2) –– skipskip3.2 Sequences and Summations

• E.g., a postulated enumeration of the reals:

r1 = 0 3 0 1 9 4 8 5 7 1 r1 0.3 0 1 9 4 8 5 7 1 …

r2 = 0.1 0 3 9 1 8 4 8 1 …

r3 = 0.0 3 4 1 9 4 1 9 3 …3

r4 = 0.9 1 8 2 3 7 4 6 1 …

……

• OK, now let’s make r’ by replacing dii by the rule.

(Rule: r’ = 0 d1 d2 d3 d4 where di = 4 if dii 4 and di = 5 if dii = 4)(Rule: r 0.d1 d2 d3 d4 … where di 4 if dii 4 and di 5 if dii 4)

• r’ = 0.4454… can’t be on the list anywhere!

• This means that the assumption({ri} is countable) is wrong,

and thus, [0,1), {ri}, is uncountable.


Documents

16. Sequences and Summations - Kangwoncs.kangwon.ac.kr/~ysmoon/courses/2011_1/dm/16.pdf · 2016-06-02 · Introduction 3.2 Sequences and Summations A sequence or series is just like