Uversty o Hw ICS: Dscrete Mthemtcs or Computer Scece I Dept. Iormto & Computer Sc., Uversty o Hw J Stelovsy bsed o sldes by Dr. Be d Dr. Stll Orgls by Dr. M. P. Fr d Dr. J.L. Gross Provded by McGrw-Hll ICS : Dscrete Mthemtcs I Fll -
Uversty o Hw Lecture Chpter. Bsc Structures. Sequeces d Summtos ICS : Dscrete Mthemtcs I Fll -
Summto Notto Uversty o Hw Gve sequece { }, teger lower boud or lmt, d teger upper boud, the the summto o { } rom to s wrtte d deed s ollows:... Here, s clled the dex o summto. m m l l ICS : Dscrete Mthemtcs I Fll -
Geerlzed Summtos Uversty o Hw For te sequece, we wrte: To sum ucto over ll members o set X {x, x, }: x X x x x Or, X {x Px}, we my ust wrte: x x x P x ICS : Dscrete Mthemtcs I Fll -
Smple Summto Exmple Uversty o Hw 9 6 5 7 ICS : Dscrete Mthemtcs I Fll -5
-6 ICS : Dscrete Mthemtcs I Fll Uversty o Hw More Summto Exmples A te sequece wth te sum: Usg predcte to dee set o elemets to sum over: 87 9 5 9 7 5 < x x x s prme
-7 ICS : Dscrete Mthemtcs I Fll Uversty o Hw 8 6 Summto Mpultos Some hdy dettes or summtos: Summg costt vlue c c Number o terms the summto
-8 ICS : Dscrete Mthemtcs I Fll Uversty o Hw Summto Mpultos Dstrbutve lw c c
-9 ICS : Dscrete Mthemtcs I Fll Uversty o Hw Summto Mpultos A pplcto o commuttvty g g
- ICS : Dscrete Mthemtcs I Fll Uversty o Hw Idex Shtg m m Let, the 6 6 5
- ICS : Dscrete Mthemtcs I Fll Uversty o Hw More Summto Mpultos Sequece splttg m m m <
- ICS : Dscrete Mthemtcs I Fll Uversty o Hw More Summto Mpultos Order reversl
Exmple: Geometrc Progresso Uversty o Hw A geometrc progresso s sequece o the orm,,,,,, where,r R. The sum o such sequece s gve by: S We c reduce ths to closed orm v clever mpulto o summtos... ICS : Dscrete Mthemtcs I Fll -
- ICS : Dscrete Mthemtcs I Fll Uversty o Hw Here we go... Geometrc Sum Dervto... r r r r rs S
-5 ICS : Dscrete Mthemtcs I Fll Uversty o Hw Dervto Exmple Cot... r S r rs
-6 ICS : Dscrete Mthemtcs I Fll Uversty o Hw Cocludg Log Dervto... S r r r r S r r S r S rs r S rs, Whe whe
Exmple: Impress Your Freds Uversty o Hw Bost, I m so smt; gve me y -dgt umber, d I ll dd ll the umbers rom to my hed ust ew secods. I.e., Evlute the summto: There s smple closed-orm ormul or the result, dscovered by Guss t ge! Ad requetly redscovered by my ICS : Dscrete Mthemtcs I Fll -7
Guss Trc, Illustrted Uversty o Hw Cosder the sum: // - We hve / ps o elemets, ech p summg to, or totl o /. ICS : Dscrete Mthemtcs I Fll -8
-9 ICS : Dscrete Mthemtcs I Fll Uversty o Hw Symbolc Dervto o Trc... l l l l l l For cse where s eve /
- ICS : Dscrete Mthemtcs I Fll Uversty o Hw Cocludg Guss Dervto So, you oly hve to do esy multplcto your hed, the cut hl. Also wors or odd prove ths t home.
Some Shortcut Expressos Uversty o Hw Geometrc sequece Guss trc Qudrtc seres Cubc seres ICS : Dscrete Mthemtcs I Fll -
Usg the Shortcuts Uversty o Hw Exmple: Evlute 5 Use seres splttg. Solve or desred summto. Apply qudrtc seres rule. Evlute. ICS : Dscrete Mthemtcs I Fll 5 9 97,95. 5 9 9 5 99 6 6 8,5,5 -
Cdlty Uversty o Hw The sets A d B hve the sme cdlty d oly there s oe-to-oe correspodece rom A to B. A set tht s ether te or hs the sme cdlty s the set o postve tegers s clled coutble. A set tht s ot coutble s clled ucoutble. Exmple: Show tht the set o odd postve tegers s coutble set. Cosder the ucto rom Z to the set o odd postve tegers ICS : Dscrete Mthemtcs I Fll A oe-to-oe correspodece betwee Z d the set o odd postve tegers. -
Cdlty cot. Uversty o Hw A te set S s coutble t s possble to lst the elemets o the set sequece dexed by the postve tegers,,,, s oeto-oe mppg : Z S where,,,, Exmple: Show tht the set o postve rtol umbers s coutble see gure ICS : Dscrete Mthemtcs I Fll -
-5 ICS : Dscrete Mthemtcs I Fll Uversty o Hw Summto Mpultos Useul dettes: < m m m Groupg. Order reversl. Sequece splttg.