Sequences & Summations
CS 1050
Rosen 1.7
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Sequences and Summations: Understanding the Basics of Discrete Structures, Study notes of Computer Science
An introduction to sequences and summations, two fundamental concepts in discrete mathematics. A sequence is an ordered list of terms, while summations allow us to calculate the sum of terms in a sequence. Notations, common sequences, arithmetic summations, telescoping sums, closed form solutions, and double summations.
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Sequences & Summations
CS 1050
Rosen 1.
Sequence
A sequence is a discrete structure used to representan ordered list.
A sequence is a function from a subset of the setof integers (usually either the set {0,1,2,.. .} or{1,2, 3,.. .}to a set S.
We use the notation a
n
to denote the image of the
integer n. We call a
n
a term of the sequence.
Notation to represent sequence is {a
n
}
Common Sequences
a, a+d, a+2d, a+3d, a+4d, …1, 4, 9, 16, 25,.. .1, 8, 27, 64, 125,.. .1, 16, 81, 256, 625,.. .2, 4, 8, 16, 32,.. .3, 9, 27, 81, 243,.. .1, 2, 6, 24, 120,...
3 n!
n
2
n
n
4
n
3
n
2
Arithmetic
Summations
Notation for describing the sum of the termsa
m
, a
m+
,.. ., a
n
from the sequence, {a
n
}
n
a
m
+a
m+
+... + a
n
=
∑
a
j
j=m
j is the index of summation (dummy variable)
The index of summation runs through all integersfrom its lower limit, m, to its upper limit, n.
Summations follow all the rules
of multiplication and addition!
c
j
cj
j
n
j
n
=
=
=
=
1
1
c(1+2+…+n) = c + 2c +…+ nc
r
ar
ar
ar
ar
ar
ar
a
ar
j
j
n
j
j
n
k
n k
n
k
k
n
n
k
k
n
=
=
=
=
=
0
1
0
1 1
1
1
1
0
Telescoping Sums
a
a
a
a
a
a
a
a
a
a
a
a
j
j
n
j
n
n
n
=
−
−
∑
1
1
1
0
2
1
3
2
1
0
[
) ]
k
k
k
2
1 4
2
2
2
2
2
2
2
2
2
2
=
∑
Example
Proof
(cont.)
[
]
k
k
k
k
n
k
n
2
1
2
1
1
2
1
=
=
−
−
(
)
=
−
(
)
[
]
k
k
k
k
n
k
n
k
n
2
1
2
1
1
1
2
1
=
=
=
−
−
(
)
=
− (
)
n
k
n
k
n
2
2
1
0
2
−
=
- −
=
( )
n
n
k
k
n
2
1
2
=
= ∑
( )
k
n
n
k
n^ = ∑
=
1
2
2
Closed Form Solutions to Sums
j
n
n n
j
n
=
=
= ∑
0
1
1
2
0
...
(
) /
j
n
n n
n
j
n
2
0
2
2
2
0
1
1
2
1
6
= ∑
=
=
...
(
)(
) /
k
n
n
k
n
3
1
2
2
1
4
= ∑
=
ar
ar
a
r
r
k
k
n
n
=
∑
=
−
−
≠
0
1
1
1
,
Cardinality
Earlier we defined cardinality of a set as thenumber of elements in the set. We can extend thisidea to infinite sets.
The sets A and B have the same cardinality if andonly if there is a one-to-one correspondence fromA to B.
A set that is either finite or has the samecardinality as the set of natural numbers is called countable.
A set that is not countable is called
uncountable.