2.4 - Sequences and Series

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1 2.4 - Sequeces ad Series Sequeces A sequece is a ordered list of elemets. Defiitio 1 A sequece is a fuctio from a subset of the set of itegers (usually either the set 80, 1, 2, 3,... < or the set 81, 2, 3,... <) to a set S. We use the otatio a to deote the image of the iteger. We call a a term of the sequece. We use 8a < to describe the sequece 8a 1, a 2, a 3,...<. ü Example What are the terms a 0, a 1, a 2, ad a 3 of the sequece give by a = 2 + H-2L? There are two particular types of sequeces you have see before: geometric ad arithmetic. Defiitio 2 A geometric progressio is a sequece of the form a, ar, ar 2,..., ar,... where the iitial term a ad the commo ratio r are real umbers. A geometric progressio is give by the expoetial fuctio f HL = ar. Defiitio 3 A arithmetic progressio is a sequece of the form a, a + d, a + 2 d,..., a + d,... where the iitial term a ad the commo differece d are real umbers. A arithmetic progressio is give by the liear fuctio f HL = dx+ a. ü Example Idetify the followig sequeces as arithmetic, geometric, or either. If the sequece is arithmetic, idetify the iitial term ad commo differece. If the sequece is geometric, idetify the iitial term ad commo ratio. (a) a = 12 I- 1 3 M (b) a = 12-7

2 2 Lecture_02_04.b Fiite sequeces of the form a 1, a 2, a 3,..., a (or a 1 a 2 a 3 a ) are used i computer sciece ad are called strigs. The umber of terms i a strig S is called the legth of the strig. A strig with o terms is called a empty strig ad is deoted by l, ad has a legth of zero. Recurrece Relatios So far we have cosidered cases where we have a explicit formula (or closed formula) for geeratig terms of a sequece. We ca also specify a formula usig a recurrece relatio. Defiitio 4 A recurrece relatio for the sequece 8a < is a equatio that expresses a i terms of oe or more of the previous terms of the sequece, amely, a 0, a 1,..., a -1, for all itegers with 0, where 0 is a oegative iteger. A sequece is called a solutio of the recurrece relatio if its terms satisfy the recurrece relatio. (A recurrece relatio is said to recursively defie a sequece.) ü Example Let 8a < be a sequece that satisfies the recurrece relatio a = 2 a -1 for = 1, 2, 3, 4,... ad suppose a 0 = 3. What are a 1, a 2, a 3,ad a 4? Oe particularly well kow recurrece relatio is give below. Defiitio 5 The Fiboacci sequece, f 0, f 1, f 2,..., is defied by the iitial coditios f 0 = 0, f 1 = 1, ad the recurrece relatio f = f -1 + f -2 for = 2, 3, 4,... ü Example Fid the Fiboacci umbers f 2, f 3, f 4, ad f 5. Special Iteger Sequeces A importat problem is fidig a formula or geeral rule for geerate the terms of the sequece. Whe tryig to fid a formula for a sequece, several questios to cosider. Are there rus of the same value? That is, does the same value occur may times i a row? Are terms obtaied from previous terms by addig the same amout, or a amout that depeds o the positio i the sequece? Are terms obtaied from previous terms by multiplyig by a particular amout? Are terms obtaied by combiig previous terms i a certai way? Are there cycles amog terms?

3 Lecture_02_04.b 3 The followig is a list of useful sequeces. Table 1 Some Useful Sequeces th Term First Te Terms 2 1, 4, 9, 1, 25, 3, 49, 4, 81, 100, , 8, 27, 4, 125, 21, 343, 512, 729, 1000, , 1, 81, 25, 25, 129, 2401, 409, 51, , , 4, 8, 1, 32, 4, 128, 25, 512, 1024, , 9, 27, 81, 729, 2187, 51, 19 83, ,...! 1, 2,, 24, 120, 720, 5040, , , ,... f 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,... ü Example For each of these lists of itegers, provide a simple formula or rule that geerates the terms of a iteger sequece that begis with the give list. Assumig that your formula or rule is correct, determie the ext three terms of the sequece. (a) 3,, 11, 18, 27, 38, 51,, 83, 102,... (b) 2, 4, 1, 25, 5 53, ,.... Oe useful website for fidig a formula for iteger sequeces is the Ecyclopedia of Iteger Sequeces ( Summatios Give a sequece a 1, a 2,..., a or say a m, a m+1,..., a, we use summatio otatio to express the sum of the terms. So we write a 1 + a a = a j or a m + a m a = a j j=m where j is the idex of summatio (the choice of j is arbitrary, some use i or k), 1 or m is the lower limit ad is the upper limit. Recall that if a ad b are real umbers, the Iaxj + by j M = a x j + b where x 1, x 2,..., x ad y 1, y 2,..., y are real umbers. b j

4 4 Lecture_02_04.b ü Example 2.4. Evaluate the followig sums. 4 (a) 2 H-3L j j=0 (b) H4 j + 1L j=3 Occasioally we will have to rewrite a sum so the idex starts at 0 or 1. ü Example Rewrite the sum so that j starts at zero. H4 j + 1L. j=3 We ca also use summatio otatio to add all values of a fuctio, or terms of a idexed set, where the idex of summatio rus over all values i a give set. I this case we would write f HsL to represet the sum of the values sœs f HsL for all members s œ S. ü Example What is the value of H4 j + 1L where S = 82, 3, 5, 7<? jœs The sum i part (b) of the Example 2.4. is a example of a arithmetic series, ad the sum i part (a) is a example of a geometric series. Recall that there are formulas to help evaluate the sum of a geometric series.

5 Lecture_02_04.b 5 Theorem 1 If a ad r are real umbers ad r 0, the j=0 ar j = ar +1 - a r - 1 if r 1. H + 1L a if r = 1 ü Example Evaluate the followig sum 2 H-3L j. j=0 We ca also have double ad triple sums. ü Example Evaluate the followig sum Ii 2 + j 3 M. i=0 j=0 As we metioed earlier, i some cases, we have shortcut formulas to help evaluate certai types of sums. Table 2 Some Useful Summatio Formulas Sum Closed Form ar k ar Hr 0L + 1 a, r 1 r 1 k=0 H + 1L k 2 k 2 H + 1L H2+1L k 3 2 H + 1L 2 x k, x < 1 kx k 1, x < 1 ü Example Evaluate the followig sum k 3. j= x 4 1 H1 xl 2 ü

Sequences, Sums, and Products

Sequences, Sums, and Products CSCE 222 Discrete Structures for Computig Sequeces, Sums, ad Products Dr. Philip C. Ritchey Sequeces A sequece is a fuctio from a subset of the itegers to a set S. A discrete structure used to represet