CS 441 Discrete Mathematics for CS Lecture 10

Sequences and summations

Milos Hauskrecht [email protected] 5329 Sennott Square

CS 441 Discrete mathematics for CS M. Hauskrecht

Sequences

Definition: A sequence is a function from a subset of the set of integers (typically the set {0,1,2,...} or the set {1,2,3,...} to a set

S. We use the notation an to denote the image of the integer n. We call an a term of the sequence.

Notation: {an} is used to represent the sequence (note {} is the same notation used for sets, so be careful). {an} represents the ordered list a1, a2, a3, ... .

1 2 3 4 5 6 ….

a1 a2 a3 a4 a5 a6 ….

{an}

CS 441 Discrete mathematics for CS M. Hauskrecht

1 Sequences

Examples: 2 • (1) an = n , where n = 1,2,3... – What are the elements of the sequence? 1, 4, 9, 16, 25, ... n • (2) an = (-1) , where n=0,1,2,3,... – Elements of the sequence? 1, -1, 1, -1, 1, ... n • 3) an = 2 , where n=0,1,2,3,... – Elements of the sequence? 1, 2, 4, 8, 16, 32, ...

CS 441 Discrete mathematics for CS M. Hauskrecht

Arithmetic progression

Definition: An arithmetic progression is a sequence of the form a, a+d,a+2d, …, a+nd where a is the initial term and d is common difference, such that both belong to R.

Example:

•sn= -1+4n for n=0,1,2,3, … • members: -1, 3, 7, 11, …

CS 441 Discrete mathematics for CS M. Hauskrecht

Definition A geometric progression is a sequence of the form: a, ar, ar2, ..., ark, where a is the initial term, and r is the common ratio. Both a and r belong to R.

Example: n •an = ( ½ ) for n = 0,1,2,3, … members: 1,½, ¼, 1/8, …..

CS 441 Discrete mathematics for CS M. Hauskrecht

Sequences

• Given a sequence finding a rule for generating the sequence is not always straightforward

Example: • Assume the sequence: 1,3,5,7,9, …. • What is the formula for the sequence? • Each term is obtained by adding 2 to the previous term. 1, 1+2=3, 3+2=5, 5+2=7 • What type of progression this suggest?

CS 441 Discrete mathematics for CS M. Hauskrecht

3 Sequences

• Given a sequence finding a rule for generating the sequence is not always straightforward

Example: • Assume the sequence: 1,3,5,7,9, …. • What is the formula for the sequence? • Each term is obtained by adding 2 to the previous term. • 1, 1+2=3, 3+2=5, 5+2=7 • It suggests an arithmetic progression: a+nd with a=1 and d=2

•an=1+2n

CS 441 Discrete mathematics for CS M. Hauskrecht

Sequences

• Given a sequence finding a rule for generating the sequence is not always straightforward

Example 2: • Assume the sequence: 1, 1/3, 1/9, 1/27, … • What is the sequence? • The denominators are powers of 3. 1, 1/3= 1/3, (1/3)/3=1/(3*3)=1/9, (1/9)/3=1/27 • This suggests a geometric progression: ark with a=1 and r=1/3 • (1/3 )n

CS 441 Discrete mathematics for CS M. Hauskrecht

4 Recursively defined sequences

• The n-th element of the sequence {an} is defined recursively in terms of the previous elements of the sequence and the initial elements of the sequence.

Example :

•an = an-1 + 2 assuming a0 = 1;

•a0 = 1;

•a1 = 3;

•a2 = 5;

•a3 = 7;

• Can you write an non-recursively using n?

•an = 1 + 2n

CS 441 Discrete mathematics for CS M. Hauskrecht

Fibonacci sequence

• Recursively defined sequence, where

•f0 = 0;

•f1 = 1;

•fn = fn-1 + fn-2 for n = 2,3, …

•f2 = 1

•f3 = 2

•f4 = 3

•f5 = 5

CS 441 Discrete mathematics for CS M. Hauskrecht

5 Summations

Summation of the terms of a sequence: n a j a m a m 1 ... a n j m The variable j is referred to as the index of summation. • m is the lower limit and • n is the upper limit of the summation.

CS 441 Discrete mathematics for CS M. Hauskrecht

Summations

Example: • 1) Sum the first 7 terms of {n2} where n=1,2,3, ... .

• 7 7 2 a j j 1 4 16 25 36 49 140 jj1 1

• 2) What is the value of 8 8 j a j ( 1) 1 ( 1) 1 ( 1) 1 1 kk4 4

CS 441 Discrete mathematics for CS M. Hauskrecht

6 Arithmetic series

Definition: The sum of the terms of the arithmetic progression a, a+d,a+2d, …, a+nd is called an arithmetic series.

Theorem: The sum of the terms of the arithmetic progression a, a+d,a+2d, …, a+nd is n n n(n 1) S (a jd) na d j na d j11j 2

• Why?

CS 441 Discrete mathematics for CS M. Hauskrecht

Arithmetic series

Theorem: The sum of the terms of the arithmetic progression a, a+d,a+2d, …, a+nd is n n n(n 1) S (a jd) na d j na d j11j 2 Proof: n n n n S (a jd) a jd na d j j11j1 j j1

n j 1 2 3 4 .... (n 2) (n 1) n j1

CS 441 Discrete mathematics for CS M. Hauskrecht

7 Arithmetic series

Theorem: The sum of the terms of the arithmetic progression a, a+d,a+2d, …, a+nd is

n n n(n 1) S (a jd) na d j na d j11j 2 Proof: n n n n S (a jd) a jd na d j j11j1 j j1

n j 1 2 3 4 .... (n 2) (n 1) n j1

n+1 n+1 … n+1 n *(n 1) 2 CS 441 Discrete mathematics for CS M. Hauskrecht

Arithmetic series 5 Example: S (2 j3) j1 5 5 2 j3 jj1 1

5 5 21 3 j jj1 1

5 2*5 3 j j1 (5 1) 10 3 *5 2

10 45 55

CS 441 Discrete mathematics for CS M. Hauskrecht

8 Arithmetic series

5 Example 2: S (2 j3) j3 5 2 (2 j3) (2 j3) Trick j1 j1 5 5 2 2 21 3 j 21 3 j jj1 1 jj1 1

5513 42

CS 441 Discrete mathematics for CS M. Hauskrecht

Double summations 4 2 Example: S (2i j) ij1 1 4 2 2 2i j ij1 1 j 1 4 2 2 2i 1 j ijj1 1 1 4 2 2i*2 j ij1 1 4 2i*2 3 i1 4 4 4i 3 ii1 1 4 4 4i 3 1 4*10 3*4 28 ii1 1

CS 441 Discrete mathematics for CS M. Hauskrecht

Definition: The sum of the terms of a geometric progression a, ar, ar2, ..., ark is called a geometric series.

Theorem: The sum of the terms of a geometric progression a, ar, ar2, ..., arn is

n n n1 j j r 1 S (ar ) a r a j00j r 1

CS 441 Discrete mathematics for CS M. Hauskrecht

Geometric series

Theorem: The sum of the terms of a geometric progression a, ar, 2 n ar , ..., ar is n n n1 j j r 1 S (ar ) a r a j00j r 1 Proof: n S ar j a ar ar 2 ar 3 ... ar n j0 • multiply S by r n rS rar j ar ar 2 ar 3 ... ar n1 j0 • Substract rS S ar ar 2 ar 3 ... ar n1 a ar ar 2.. ar n ar n1 a ar n1 a r n1 1 S a r 1 r 1

CS 441 Discrete mathematics for CS M. Hauskrecht

10 Geometric series

Example: 3 S 2(5) j j0

General formula: n n n1 j j r 1 S (ar ) a r a j00j r 1

3 54 1 S 2(5) j 2* j0 5 1 625 1 624 2* 2* 2*156 312 4 4

CS 441 Discrete mathematics for CS M. Hauskrecht

Infinite geometric series

• Infinite geometric series can be computed in the closed form for x<1 • How?

k k 1 n n x 1 1 1 x lim k x lim k n0 n0 x 1 x 1 1 x

• Thus: 1 x n n 0 1 x

CS 441 Discrete mathematics for CS M. Hauskrecht

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