(Sequences and Summations) - Kangwoncs.kangwon.ac.kr/~ysmoon/courses/2006_1/dm/32. Sequences... · 2016-06-02 · 1 이산수학(Discrete Mathematics) 3.2 수열과합 (Sequences

1

이산수학 (Discrete Mathematics)

3.2 수열과 합

(Sequences and Summations)

20062006년년 봄학기봄학기

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강원대학교강원대학교 컴퓨터과학과컴퓨터과학과

Page 2Discrete Mathematicsby Yang-Sae Moon

IntroductionIntroduction3.2 Sequences and Summations

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

• 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 terms in a (possibly infinite) series. (∑)

2


SequencesSequences3.2 Sequences and Summations

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

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

The index of an is n. (Or, often i is used.)

S A1234::

a1 = f(1)a2 = f(2)a3 = f(3)a4 = f(4)

::

f


Sequence ExamplesSequence Examples3.2 Sequences and Summations

Example of an infinite series (무한 수열)

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

• Then, {an} = 1, 1/2, 1/3, 1/4, …

Example with repetitions (반복 수열)

• 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, not the 2-element set {1, −1}.

3


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.(순열의 몇몇 값들에 기반하여 f(n)을 발견하는 문제에 자주 직면하게 된다.)

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

• 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 subsequence.(세상에는 시퀀스를 생성하는 셀 수 없이 많은 함수가 존재한다.)

4


What are Strings? (1/2)What are Strings? (1/2)3.2 Sequences and Summations

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

More formally,

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

• A string s over alphabet Σ is any sequence {si} of symbols, si∈Σ, 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.


What are Strings? (2/2)What are Strings? (2/2)3.2 Sequences and Summations

More string notation

• 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 ncopies of s. (스트링 s를 n번 concatenation하는 표현)

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

• If Σ is an alphabet and n∈N,

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

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

5


Summation NotationSummation Notation3.2 Sequences and Summations

Given a sequence {an}, an integer lower bound j≥0, and an

integer upper bound k≥j, then the summation of {an} from

j to k is written and defined as follows:({an}의 i번째에서 j번째까지의 합, 즉, aj로부터 ak까지의 합)

Here, i is called the index of summation.

kjj

k

ji i

k

jii a...aa:aa +++≡= +=

=∑∑ 1


Generalized SummationsGeneralized Summations3.2 Sequences and Summations

For an infinite series, we may write:

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

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

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

++≡∑∈

21

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

++≡∑ 21

...aa:a jjji

i ++≡ +

∞

=∑ 1

6


Summation ExamplesSummation Examples3.2 Sequences and Summations

A Simple example

An infinite sequence with a finite sum:

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

3217105

)116()19()14(

)14()13()12(1 2224

2

2

=++=

+++++=

+++++=+∑=i

i

21222 41

2110

0

=+++=++= −∞

=

−∑ ......i

i

874925947532x 2222

10) prime is (

2 =+++=+++=∑<∧ xx


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

Some useful identities for summations:

∑∑

∑ ∑∑

∑∑

+

+==

−=

+⎟⎠

⎞⎜⎝

⎛=+

=

nk

nji

k

ji

x xx

xx

)ni(f)i(f

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

)x(fc)x(cf (Distributive law)

(Application ofcommutativity)

(Index shifting)

7


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

Some more useful identities for summations:

∑∑

∑∑

∑∑∑

==

−

==

+===

++=

−=

<≤+⎟⎟⎠

⎞⎜⎜⎝

⎛=

k

i

k

i

jk

i

k

ji

k

mi

m

ji

k

ji

)i(f)i(f)i(f

)ik(f)i(f

kmj)i(f)i(f)i(f

0

2

0

0

1

122

if

(Grouping)

(Order reversal)

(Series splitting)


An Interesting ExampleAn Interesting Example3.2 Sequences and Summations

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

all the numbers from 1 to n in my head in just a few

seconds.” (1에서 n까지의 합을 수초 내에 계산하겠다!)

I.e., Evaluate the summation:

There is a simple formula for the result, discovered by

Euler at age 12!

∑=

n

i

i1

8


EulerEuler’’s Trick, Illustrateds Trick, Illustrated3.2 Sequences and Summations

Consider the sum:

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

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.)

…

n+1n+1

n+1

21

1

)n(ni

n

i

+=∑

=


Symbolic Derivation of Trick (1/2)Symbolic Derivation of Trick (1/2)3.2 Sequences and Summations

...)in(i)in(i

))i(n(i)in(i

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

))k(i(iiiii

k

i

k

i

kn

i

k

i

kn

i

k

i

)k(n

i

k

i

)k(n

i

k

i

)k(n

i

k

i

n

ki

k

i

k

i

n

i

11

1

11

1

1111

11

1

01

1

01

1

0111

2

11

=−++⎟⎠

⎞⎜⎝

⎛=−++⎟

⎠

⎞⎜⎝

⎛=

−−+⎟⎠

⎞⎜⎝

⎛=−+⎟

⎠

⎞⎜⎝

⎛=

++−+−+⎟⎠

⎞⎜⎝

⎛=

+++⎟⎠

⎞⎜⎝

⎛=+⎟

⎠

⎞⎜⎝

⎛==

∑∑∑∑

∑∑∑∑

∑∑

∑∑∑∑∑∑

==

−

==

−

==

+−

==

+−

==

+−

==+====

∑∑

∑∑

==

−

==

−=⇒

−=

k

i

k

i

jk

i

k

ji

)ik(f)i(f

)ik(f)i(f

00

0

kn 2 since =

9


Symbolic Derivation of Trick (2/2)Symbolic Derivation of Trick (2/2)3.2 Sequences and Summations

21

111

1 1

21

1111

/)n(n

)n()n(k)n(

)ini()in(ii

nk

i

k

i

k

i

k

i

n

i

+=

+=+=+=

−++=−++⎟⎠

⎞⎜⎝

⎛=

∑

∑∑∑∑

=

====

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,

ar3, …, ark, where a,r∈R.

The sum of such a sequence is given by:

We can reduce this to closed form via clever manipulation

of summations...

∑=

=k

i

iarS0

10


Derivation of Geometric Sum (1/3)Derivation of Geometric Sum (1/3)3.2 Sequences and Summations

...arararar

ararar

rararrrararrrS

arS

nn

i

in

ni

in

i

i

n

i

in

i

)i(n

i

i

n

i

in

i

in

i

in

i

i

n

i

i

=+⎟⎠

⎞⎜⎝

⎛=+⎟

⎠

⎞⎜⎝

⎛=

===

====

=

+

=

+

+==

+

=

+

=

−+

=

+

====

=

∑∑∑

∑∑∑

∑∑∑∑

∑

1

1

1

11

1

1

1

1

11

0

1

0

1

000

0



)r(aS)r(aar

aararar

arararar

arar)arar(ararrS

nnn

i

i

nn

i

i

i

i

nn

i

i

nn

i

inn

i

i

11 11

0

1

1

0

0

01

1

0

1

1

001

1

−+=−+⎟⎠

⎞⎜⎝

⎛=

−+⎟⎠

⎞⎜⎝

⎛+⎟

⎠

⎞⎜⎝

⎛=

−+⎟⎠

⎞⎜⎝

⎛+=

+⎟⎠

⎞⎜⎝

⎛+−=+⎟

⎠

⎞⎜⎝

⎛=

++

=

+

==

+

=

+

=

+

=

∑

∑∑

∑

∑∑

11



a)n(aaarSr

rr

raS

)r(a)r(S

)r(aSrS

)r(aSrS

n

i

n

i

in

i

i

n

n

n

n

111 ,1 when

1 en wh11

11

1

1

000

1

1

1

1

+=⋅====

≠⎟⎟⎠

⎞⎜⎜⎝

⎛−−

=

−=−

−=−

−+=

∑∑∑===

+

+

+

+


Nested SummationsNested Summations3.2 Sequences and Summations

These have the meaning you’d expect.

( )

60106

4321666

321

4

1

4

1

4

1

4

1

3

1

4

1

3

1

4

1

3

1

=⋅=

+++===

++=⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛=

∑∑

∑∑ ∑∑ ∑∑∑

==

== == == =

)(ii

ijiijij

ii

ii ji ji j

12


Some Shortcut ExpressionsSome Shortcut Expressions3.2 Sequences and Summations

∑=

n

k

k1

Closed FormSum

10

≠∑=

r,arn

k

k

∑=

n

k

k1

3

∑=

n

k

k1

2

10

<∑∞

=

x,xk

k

)r()r(a n

111

−−+

21)n(n +

6121 )n)(n(n ++

41 22 )n(n +

11

1 <∑∞

=

− x,kxk

k

x−11

211

)x( −

Infinite series(무한급수)


Infinite Series (Infinite Series (무한급수무한급수) (1/2)) (1/2)3.2 Sequences and Summations

• Let a = 1 and r = x, then

• If k ∞, then xk+1 0

• Therefore,

10

<∑∞

=x,x

nn

11

0

1

∑ =

+

−−

=k

n

kn

xx

kx

⎟⎟⎠

⎞⎜⎜⎝

⎛−−

=∑ =

+n

j

kj

raar

ar0

1

1 since

xxx

nn

−=

−−

=∑∞

= 11

1

10

13


Infinite Series (Infinite Series (무한급수무한급수) (2/2)) (2/2)3.2 Sequences and Summations

10

1 <∑∞

=− x,kx

nn

xx

nn

−=∑∞

= 11

0

211

11

)x(nx

nn

−=⇒ ∑∞

=−

xdxd

xdxd

nn

−=⇒ ∑∞

= 11

0

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−=

= −

2

1

recall

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

)x(fdxd

nxxdxd nn


Using the ShortcutsUsing the Shortcuts3.2 Sequences and Summations

Example: Evaluate .

• Use series splitting.

• Solve for desired

summation.

• Apply quadratic

series rule.

• Evaluate.

∑=

100

50

2

k

k

.,,,

kkk

kkk

kkk

kkk

92529742540350338

6995049

6201101100

49

1

2100

1

2100

50

2

100

50

249

1

2100

1

2

=−=

⋅⋅−

⋅⋅=

−⎟⎠

⎞⎜⎝

⎛=

+⎟⎠

⎞⎜⎝

⎛=

∑∑∑

∑∑∑

===

===

14


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

there exists a bijection (bijective function) from A to B.(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

elements n∈N.(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 uncountable.(유한집합이거나, 자연수 집합과 크기가 동일하면 countable하며, 그렇지 않으

면 uncountable하다.)

Intuition behind “countable:” we can enumerate

(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.)

15


Countable Sets: ExamplesCountable Sets: Examples3.2 Sequences and Summations

Theorem: The set Z is countable.

• Proof: Consider f:Z→N where f(i)=2i for i≥0 and f(i) = −2i−1 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)

(1,3)

(1,4)

(1,5)

(2,1)

(2,2)

(2,3)

(2,4)

(2,5)

(3,1)

(3,2)

(3,3)

(3,4)

(3,5)

(4,1)

(4,2)

(4,3)

(4,4)

(4,5)

(5,1)

(5,2)

(5,3)

(5,4)

(5,5)

… … … … …

…

…

…

…

…

…

consider sum is 2, thenconsider sum is 3, thenconsider sum is 4, thenconsider sum is 5, thenconsider sum is 6, thenconsider …

Note a set of rational numbers is countable!


Uncountable Sets: Example (1/2)Uncountable Sets: Example (1/2)3.2 Sequences and Summations

Theorem: The open interval

[0,1) :≡ {r∈R| 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…

r4 = 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.

16


Uncountable Sets: Example (2/2)Uncountable Sets: Example (2/2)3.2 Sequences and Summations

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

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 …

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)

• 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.

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(Sequences and Summations) - Kangwoncs.kangwon.ac.kr/~ysmoon/courses/2006_1/dm/32. Sequences... · 2016-06-02 · 1 이산수학(Discrete Mathematics) 3.2 수열과합 (Sequences