Transcript Sequences & Summations CS 1050 Rosen 3.2 Sequence • A sequence is a discrete structure used to represent an ordered list. • A sequence is.

Sequences & Summations
CS 1050
Rosen 3.2
Sequence
• A sequence is a discrete structure used to represent
an ordered list.
• A sequence is a function from a subset of the set
of integers (usually either the set {0,1,2,. . .} or
{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 to represent sequence is {an}
Examples
• {1, 1/2, 1/3, 1/4, . . .} or the sequence {an}
where an = 1/n, nZ+ .
• {1,2,4,8,16, . . .} = {an} where an = 2n, nN.
• {12,22,32,42,. . .} = {an} where an = n2, nZ+
Common Sequences
Arithmetic
a, a+d, a+2d, a+3d, a+4d, …
n2
1, 4, 9, 16, 25, . . .
n3
1, 8, 27, 64, 125, . . .
n4
1, 16, 81, 256, 625, . . .
2n
2, 4, 8, 16, 32, . . .
3n
3, 9, 27, 81, 243, . . .
n!
1, 2, 6, 24, 120, . . .
Summations
• Notation for describing the sum of the terms
am, am+1, . . ., an from the sequence, {an}
n
am+am+1+ . . . + an =  aj
j=m
• j is the index of summation (dummy variable)
• The index of summation runs through all integers
from its lower limit, m, to its upper limit, n.
Examples
5
4
j 1
j 0
 j  ( j  1)  1 2  3  4  5  15
5
1
j 11 2  1 3  1 4  1 5
j 1
5
1  1 j  1 1 2  1 3  1 4  1 5
j2
Summations follow all the rules
of multiplication and addition!
n
n
j 1
j 1
c  j   cj  c(1+2+…+n) = c + 2c +…+ nc
n
n
r  ar   ar
j
j 0
j 0
j 1
n1
  ar 
k
k 1
n
ar
n1
  ar  ar
k
k 1
n
n1
 a   ar
k 0
k
Telescoping Sums
n
 (a
j
 a j 1 )  (a1  a0 )  (a2  a1 ) 
j 1
(a3  a2 )  ... (an  an 1 )  an  a0
Example
4
2
2
[k

(k

1)
]

k 1
(1  0 )  (2  1 )  (3  2 )  (4  3 )
2
2
2
4  16  0  16
2
2
2
2
2
2
Closed Form Solutions
A simple formula that can be used to calculate a sum without
doing all the additions.
Example:
n(n  1)
k 2
k 1
n
Proof: First we note that k2 - (k-1)2 = k2 - (k2-2k+1) = 2k-1.
Since k2-(k-1)2 = 2k-1, then we can sum each side from k=1 to
k=n
n
n
 [k
k 1
2
 k  1 ]   2k  1
2
k 1
Proof (cont.)
n
 [k
2
k 1
n
 [k
k 1
2
n
 k  1 ]   2k  1
2
k 1
n
n
k 1
k 1
 k  1 ]   2k  1
2
n
n  0  2 (k )  n
2
2
k 1
n
n 2  n  2 (k)
k 1
n2  n
k 2
k 1
n
Closed Form Solutions to Sums
n
 j  0  1  ...  n  n(n  1)/ 2
j 0
n
j
2
 0  1  ... n  n(n  1)(2n  1)/ 6
2
2
2
j 0
n n  1
k  4
k 1
n
2
2
3
n1
ar
a
k
 ar  r  1 ,r  1
k 0
n
Double Summations
4
 3  4
ij    ij   i  2i  3i    6i 


i 1 j 1
i 1 j 1 
i 1
i1
6  12 18  24  60
4
3
4
Cardinality
• Earlier we defined cardinality of a set as the
number of elements in the set. We can extend this
idea to infinite sets.
• 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 natural numbers is called
countable. A set that is not countable is called
uncountable.
Cardinality
• Cardinality of set of natural numbers?
• An infinite set is countable if and only if it is
possible to list the elements in a sequence
(indexed by the positive integers).
– Why? A one-to-one correspondence f can be expressed
in terms of the sequence a1, a2, a3…., where a1 = f(1),
a2 =f(2), etc.
– One-to-one correspondence for set of odd positive
integers (in terms of positive integers)?
f(n) = 2n - 1