Sequences & Summa,ons

Transcription

1 Sequeces & Summa,os Sec,o 2.4 of Rose Sprig 2012 CSCE 235 Itroduc5o to Discrete Structures Course web- page: cse.ul.edu/~cse235 Ques,os: Piazza

2 Outlie Although you are (more or less) familiar with sequeces ad summa5os, we give a quick review Sequeces Defii5o, 2 examples Progressios: Special sequeces Geometric, arithme5c Summa5os Careful whe chagig lower/upper limits Series: Sum of the elemets of a sequece Examples, ifiite series, covergece of a geometric series 2

3 Sequeces Defii,o: A sequece is a fuc5o from a subset of itegers to a set S. We use the ota5o(s): {a } {a } {a } =0 Each a is called the th term of the sequece We rely o the cotext to dis5guish betwee a sequece ad a set, although they are dis5ct structures 3

4 Sequeces: Example 1 Cosider the sequece {(1 + 1/) } =1 The terms of the sequece are: What is this sequece? a 1 = (1 + 1/1) 1 = a 2 = (1 + 1/2) 2 = a 3 = (1 + 1/3) 3 = a 4 = (1 + 1/4) 4 = a 5 = (1 + 1/5) 5 = The sequece correspods to lim {(1 + 1/) } =1 = e =

5 Sequeces: Example 2 The sequece: {h } =1 = 1/ is kow as the harmoic sequece The sequece is simply: 1, 1/2, 1/3, 1/4, 1/5, This sequece is par5cularly iteres5g because its summa5o is diverget: Σ =1 (1/) = 5

6 Progressios: Geometric Defii,o: A geometric progressio is a sequece of the form Where: a, ar, ar 2, ar 3,, ar, a R is called the ii5al term r R is called the commo ra5o A geometric progressio is a discrete aalogue of the expoe5al fuc5o f(x) = ar x 6

7 Geometric Progressios: Examples A commo geometric progressio i Computer Sciece is: with a=1 ad r=1/2 {a }= 1/2 Give the ii5al term ad the commo ra5o of {b } with b = (- 1) {c } with c = 2(5) {d } with d = 6(1/3) 7

8 Progressios: Arithme5c Defii,o: A arithmetric progressio is a sequece of the form Where: a, a+d, a+2d, a+3d,, a+d, a R is called the ii5al term d R is called the commo differece A arithme5c progressio is a discrete aalogue of the liear fuc5o f(x) = dx+a 8

9 Arithme5c Progressios: Examples Give the ii5al term ad the commo differece of {s } with s = {t } with s = 7 3 9

10 More Examples Table 1 o Page 162 (Rose) has some useful sequeces: { 2 } =1, { 3 } =1, { 4 } =1, {2 } =1, {3 } =1, {!} =1 10

11 Outlie Although you are (more or less) familiar with sequeces ad summa5os, we give a quick review Sequeces Defii5o, 2 examples Progressios: Special sequeces Geometric, arithme5c Summa,os Careful whe chagig lower/upper limits Series: Sum of the elemets of a sequece Examples, ifiite series, covergece of a geometric series 11

12 Summa5os (1) You should be by ow familiar with the summa5o ota5o: Here Σ j=m (a j ) = a m + a m a a j is the idex of the summa5o m is the lower limit is the upper limit Oie 5mes, it is useful to chage the lower/upper limits, which ca be doe i a straighjorward maer (although we must be very careful): Σ j=1 (a j ) = Σ i=o - 1 (a i+1 ) 12

13 Summa5os (2) Some5mes we ca express a summa5o i closed form, as for geometric series Theorem: For a, r R, r 0 Σ i=0 (ar i ) = (ar +1 - a)/(r- 1) if r 1 (+1)a if r = 1 Closed form = aaly5cal expressio usig a bouded umber of well- kow fuc5os, does ot ivolved a ifiite series or use of recursio 13

14 Summa5os (3) Double summa5os oie arise whe aalyzig a algorithm Σ i=1 Σ j=1i (a j ) = a 1 + a 1 +a 2 + a 1 +a 2 +a 3 + a 1 +a 2 +a 3 + +a Summa5os ca also be idexed over elemets i a set: Σ s S f(s) Table 2 o Page 166 (Rose) has very useful summa5os. Exercises (edi5o 7 th ) are great material to prac5ce o. 14

15 Outlie Although you are (more or less) familiar with sequeces ad summa5os, we give a quick review Sequeces Defii5o, 2 examples Progressios: Special sequeces Geometric, arithme5c Summa5os Careful whe chagig lower/upper limits Series: Sum of the elemets of a sequece Examples, ifiite series, covergece of a geometric series 15

16 Series Whe we take the sum of a sequece, we get a series We have already see a closed form for geometric series Some other useful closed forms iclude the followig: Σ i=k u 1 = u- k+1, for k u Σ i=0 i = (+1)/2 Σ i=0 (i 2 ) = (+1)(2+1)/6 Σ i=0 (i k ) k+1 /(k+1) 16

17 Ifiite Series Although we will mostly deal with fiite series (i.e., a upper limit of for fixed iteger), iifiite series are also useful Cosider the followig geometric series: Σ =0 (1/2 ) = 1 + 1/2 + 1/4 + 1/8 + coverges to 2 Σ =0 (2 ) = does ot coverge However ote: Σ =0 (2 ) = (a=1,r=2) 17

18 Ifiite Series: Geometric Series I fact, we ca geeralize that fact as follows Lemma: A geometric series coverges if ad oly if the absolute value of the commo ra5o is less tha 1 Whe r <1, lim Σ i=0 (ar i ) = lim Σ i=0 (ar +1 - a)/(r- 1) = a/(1- r) 18