Sequences & Summations

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

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

Summations follow all the rules of multiplication and addition! c(1+2+…+n) = c + 2c +…+ nc

Telescoping Sums Example

Closed Form Solutions A simple formula that can be used to calculate a sum without doing all the additions. Example: 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

Proof (cont.)

Closed Form Solutions to Sums

Double Summations

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

Sequences & Summations