Sequences and Summations: Understanding the Basics of Discrete Structures, Study notes of Computer Science

Download Sequences and Summations: Understanding the Basics of Discrete Structures and more Study notes Computer Science in PDF only on Docsity! Sequences & Summations CS 1050 Rosen 1.7 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} 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 j j jj = + = + + + + = == ∑∑ ( )1 1 2 3 4 5 15 0 4 1 5 1 1 1 2 1 3 1 4 1 5 1 1 1 1 2 1 3 1 4 1 5 1 5 2 5 j j j j = + + + + + = + + + + = = ∑ ∑ Summations follow all the rules of multiplication and addition! c j cj j n j n = = == ∑∑ 11 c(1+2+É+n) = c + 2c +É+ nc r ar ar ar ar ar ar a ar j j n j j n k k n n k k n n k k n = + = = + + = + = ∑ ∑ ∑ ∑ ∑ = = = + = − + 0 1 0 1 1 1 1 1 0 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 11 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 20 1 1 2 1 6 = ∑ = + + + = + +... ( )( ) / k n n k n 3 1 2 21 4= ∑ = +( ) ar ar a r rk k n n = + ∑ = −− ≠0 1 1 1, Double Summations ij ij i i i i ji iji i =       = + +( ) = = + + + = == === = ∑∑ ∑∑∑ ∑ 1 3 1 4 1 4 1 3 1 4 1 4 2 3 6 6 12 18 24 60