Sequences and summations

Transcription

1 Lecture 0 Sequeces d summtos Istructor: Kgl Km CSE) E-ml: kkm0@kokuk.c.kr Tel. : Room : New Mleum Bldg. 0 Lb : New Egeerg Bldg. 0 All sldes re bsed o CS Dscrete Mthemtcs for Computer Scece course of the Uversty of Pttsburgh gve d opeed by Mlos Huskrecht. Next topc: Course syllbus Logc d proofs Sets Fuctos Itegers d modulr rthmetc Sequeces d summtos Coutg Probblty Reltos Grphs

2 Wht We Hve Lert Fucto - A Theorem bout Ifty - Idetty Fucto - Iverso - Composto - Addtol Fuctos Wht We Wll Ler Sequece d Summto - Smple Sequeces - Recurret Relto - More Sequeces - Summto of Elemets Sequeces

3 Sequeces Defto: A sequece s fucto from subset of the set of tegers typclly the set {0,,,...} or the set {,,,...} to set S. We use the otto to deote the mge of the teger. We cll term of the sequece. Notto: { } s used to represet the sequece ote {} s the sme otto used for sets, so be creful). { } represets the ordered lst,,, { }

4 Sequeces Exmples: ) =, where =,,... Wht re the elemets of the sequece?,, 9, 6,,... ) = -), where =0,,,,... Elemets of the sequece?, -,, -,,... ) =, where =0,,,,... Elemets of the sequece?,,, 8, 6,,... Arthmetc progresso Defto: A rthmetc progresso s sequece of the form, +d,+d,, +d where s the tl term d d s commo dfferece, such tht both belog to R. Exmple: s = -+ for =0,,,, members: -,, 7,,

5 Geometrc progresso Defto A geometrc progresso s sequece of the form:, r, r,..., r k, where s the tl term, d r s the commo rto. Both d r belog to R. Exmple: = ½ ) for = 0,,,, members:,½, ¼, /8,.. Sequeces Gve sequece fdg rule for geertg the sequece s ot lwys strghtforwrd Exmple: Assume the sequece:,,,7,9,. Wht s the formul for the sequece? Ech term s obted by ddg to the prevous term., +=, +=, +=7 Wht type of progresso ths suggest?

6 Sequeces Gve sequece fdg rule for geertg the sequece s ot lwys strghtforwrd Exmple: Assume the sequece:,,,7,9,. Wht s the formul for the sequece? Ech term s obted by ddg to the prevous term., +=, +=, +=7 It suggests rthmetc progresso: +d wth = d d= =+ Sequeces Gve sequece fdg rule for geertg the sequece s ot lwys strghtforwrd Exmple : Assume the sequece:, /, /9, /7, Wht s the sequece? The deomtors re powers of., /= /, /)/=/*)=/9, /9)/=/7 Ths suggests geometrc progresso: r k wth = d r=/ / )

7 Recursvely defed sequeces The -th elemet of the sequece { } s defed recursvely terms of the prevous elemets of the sequece d the tl elemets of the sequece. Exmple : = - + ssumg 0 = ; <- recurret relto 0 = ; <- tl codto = ; We c recursvely = ; geerte ths sequece = 7; C you wrte o-recursvely usg? = + <- closed formul Recursvely defed sequeces A recurrece relto for the sequece {} s equto tht expresses terms of oe or more of the prevous terms of the sequece, mely, 0,,...,, for ll tegers wth 0, where 0 s oegtve teger. A sequece s clled soluto of recurrece relto f ts terms stsfy the recurrece relto. A recurrece relto s sd to recursvely defe sequece.

8 Fbocc sequece Recursvely defed sequece, where f 0 = 0; f = ; f = f - + f - for =,, f = f = f = f = More Complex d Geerl Sequece We wll ler more geerl pproch to defe complex sequeces lter..

9 6 Summtos Summto of the terms of sequece: The vrble s referred to s the dex of summto. m s the lower lmt d s the upper lmt of the summto. m m m... Summtos Exmple: ) Sum the frst 7 terms of { } where =,,,.... ) Wht s the vlue of ) ) ) 8 8 k k

10 Arthmetc seres Defto: The sum of the terms of the rthmetc progresso, +d,+d,, +d s clled rthmetc seres. Theorem: The sum of the terms of the rthmetc progresso, +d,+d,, +d s S d) d ) d Why? Arthmetc seres Theorem: The sum of the terms of the rthmetc progresso, +d,+d,, +d s Proof: S d) d S d) ) d d d... ) ) 7

11 8 Arthmetc seres Theorem: The sum of the terms of the rthmetc progresso, +d,+d,, +d s Proof: d d d S ) ) ) ) ) d d d S + ) * Arthmetc seres Exmple: ) S * * ) 0 0 Wht f s odd?

12 9 Arthmetc seres Exmple : ) S ) ) Trck Double summtos Exmple: ) S * * 8 * *0

13 Geometrc seres Defto: The sum of the terms of geometrc progresso, r, r,..., r k s clled geometrc seres. Theorem: The sum of the terms of geometrc progresso, r, r,..., r s r S r ) r 0 0 r Geometrc seres Theorem: The sum of the terms of geometrc progresso, r, r,..., r s r S r ) r 0 0 r Proof: S r r r r... r multply S by r Substrct 0 rs rr 0 rs S r r r... r r r r... r r r.. r r r r S r r wht f r=? 0

14 Geometrc seres Exmple: S 0 ) Geerl formul: r S r ) r 0 0 r S ) * * * *6 Ifte geometrc seres Ifte geometrc seres c be computed the closed form for x< How? k k x x lm k x lm k x x x 0 0 Thus: x 0 x

15 Useful Closed Formule Proof? Useful Closed Formule