Sequences Summations CS 1050 Rosen 3 2 Sequence

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 = 2 n, n N. • {12, 22, 32, 42, . . . } = {an} where an = n 2, n Z+

Common Sequences Arithmetic a, a+d, a+2 d, a+3 d, a+4 d, … n 2 1, 4, 9, 16, 25, . . . n 3 1, 8, 27, 64, 125, . . . n 4 1, 16, 81, 256, 625, . . . 2 n 2, 4, 8, 16, 32, . . . 3 n 3, 9, 27, 81, 243, . . . n! 1, 2, 6, 24, 120, . . .

Summations • Notation for describing the sum of the terms am+1, . . . , an from the sequence, {an} a m, 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 + 2 c +…+ 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 k 2 - (k-1)2 = k 2 - (k 2 -2 k+1) = 2 k-1. Since k 2 -(k-1)2 = 2 k-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 a 1, a 2, a 3…. , where a 1 = f(1), a 2 =f(2), etc. – One-to-one correspondence for set of odd positive integers (in terms of positive integers)? f(n) = 2 n - 1