(Sequences and Summations) - Kangwoncs.kangwon.ac.kr/~ysmoon/courses/2006_1/dm/32. Sequences... ·...
Transcript of (Sequences and Summations) - Kangwoncs.kangwon.ac.kr/~ysmoon/courses/2006_1/dm/32. Sequences... ·...
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이산수학 (Discrete Mathematics)
3.2 수열과 합
(Sequences and Summations)
20062006년년 봄학기봄학기
문양세문양세
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. (∑)
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Page 3Discrete Mathematicsby Yang-Sae Moon
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
Page 4Discrete Mathematicsby Yang-Sae Moon
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}.
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Page 5Discrete Mathematicsby Yang-Sae Moon
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
Page 6Discrete Mathematicsby Yang-Sae Moon
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.)
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Page 7Discrete Mathematicsby Yang-Sae Moon
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.
Page 8Discrete Mathematicsby Yang-Sae Moon
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)
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Page 9Discrete Mathematicsby Yang-Sae Moon
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
Page 10Discrete Mathematicsby Yang-Sae Moon
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
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Page 11Discrete Mathematicsby Yang-Sae Moon
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
Page 12Discrete Mathematicsby Yang-Sae Moon
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)
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Page 13Discrete Mathematicsby Yang-Sae Moon
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)
Page 14Discrete Mathematicsby Yang-Sae Moon
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
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Page 15Discrete Mathematicsby Yang-Sae Moon
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
+=∑
=
Page 16Discrete Mathematicsby Yang-Sae Moon
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
Page 17Discrete Mathematicsby Yang-Sae Moon
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).
Page 18Discrete Mathematicsby Yang-Sae Moon
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
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Page 19Discrete Mathematicsby Yang-Sae Moon
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
Page 20Discrete Mathematicsby Yang-Sae Moon
Derivation of Geometric Sum (2/3)Derivation of Geometric Sum (2/3)3.2 Sequences and Summations
)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
Page 21Discrete Mathematicsby Yang-Sae Moon
Derivation of Geometric Sum (3/3)Derivation of Geometric Sum (3/3)3.2 Sequences and Summations
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
+=⋅====
≠⎟⎟⎠
⎞⎜⎜⎝
⎛−−
=
−=−
−=−
−+=
∑∑∑===
+
+
+
+
Page 22Discrete Mathematicsby Yang-Sae Moon
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
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Page 23Discrete Mathematicsby Yang-Sae Moon
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)
Page 24Discrete Mathematicsby Yang-Sae Moon
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
Page 25Discrete Mathematicsby Yang-Sae Moon
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
Page 26Discrete Mathematicsby Yang-Sae Moon
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
Page 27Discrete Mathematicsby Yang-Sae Moon
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.)
Page 28Discrete Mathematicsby Yang-Sae Moon
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
Page 29Discrete Mathematicsby Yang-Sae Moon
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!
Page 30Discrete Mathematicsby Yang-Sae Moon
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
Page 31Discrete Mathematicsby Yang-Sae Moon
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.