Showig Recursive Sequeces Coverge Itroductio My studets hve sked me bout how to prove tht recursively defied sequece coverges. Hopefully, fter redig these otes, you will be ble to tckle y such problem. Prelimiries To prove tht the recursive sequece { } coverges, we eed to show tht it is bouded d mootoic. I prctice, we either show tht is icresig d bouded bove, or decresig d bouded below. For ow, let s ssume tht we re tryig to show tht our sequece is icresig d bouded bove. { } is bouded bove mes tht there is umber M such tht M for ll. Sice there re ifiitely my differet vlues of, provig y sttemet cotiig the phrse for ll is goig to be difficult. Our strtegy is to use mthemticl iductio to prove this sttemet. We will the use our boud to show tht the sequece is icresig.
Mthemticl Iductio Imgie ldder with ifiitely my rugs. If I c show you how to get oto the bottom rug of the ldder d show you geerl procedure for gettig from y rug to the ext oe, the you c get to y level o the ldder. Here s how to get to rug 3. We c get to rug becuse we kow how to get o the ldder. We kow the geerl procedure for dvcig oe rug; so we c get from rug to rug. We c lso use tht geerl procedure for gettig to rug 3. Usig the sme rgumet, we c demostrte how to get to rug 4, rug 5 d eve rug 657. I fct, for y, we c get to rug umber. We hve just demostrted tht for ll, we c get to rug. Our strtegy for sequeces will be similr. For exmple, if we wt to show tht the is bouded bove by, we show sequece { }... 3 3 4 Sice we cot prove ifiitely idividul sttemets, we prove it for oe geerl sttemet. Tht is, we prove ) b) if the + for rbitrry. Tht is, we show tht the first term is less th two d if we c get out to d sty less th, the we c go oe step further d show tht +.
Here s how we prove ech prt: ) Sice the first term is lwys give to us, this prt is esy. b) We re tryig to prove sttemet of the form if P the Q. To prove this, we ssume tht P is true, d bsed o tht ssumptio, show tht Q is true s well. The tricky prt: we use scrtch pper to figure out wht our proof will look like the we write it up d bi the scrtch pper. Mke sure you see tht ) d b) tke together llows us to show tht 4, 5 d. I fct, we c show for y, tht. 657 Showig recursive sequece is bouded Followig., exmple, we look t the sequece, ( 6) = = +. For ow I + will give you the boud, lter I will show you how to fid this umber. Clim: For ll, 6. First the scrtch pper. b) We will ssume tht 6 d try to prove tht + 6. Workig bckwrds, strtig with wht we wt to prove, + 6 ( ) + 6 6 + 6 6 (Bigo! We get to ssume this) Officil proof: ) = 6 b) Suppose 6. The + 6 ( + 6) 6 + 6. So, if 6, the + 6 By mthemticl iductio, 6 for ll. QED QED is bbrevitio for the Lti phrse quod ert demostrdum which loosely trsltes to I rest my cse. Followig trditio, we dd it to the ed of every mthemticl proof.
Aother exmple. Cosider the sequece Clim: for ll. Scrtch pper: ) (EASY) =, ( ) + = +. for ll. 3 4 b) We get to ssume d we use lgebr to show tht +. Workig bckwrds. ( ) + + 4 + 4 6. 3 Officil Proof: ) = b) Suppose. The + 4 6 3 ( + 4) + So, if, the + By mthemticl iductio, 6 for ll. QED Showig recursive sequece is icresig We look t the sequece, ( 6) = = + gi. Remember we showed bove tht + 6 for ll. Now we will show tht the sequece is icresig. Just like the other proofs, we work bckwrds o our scrtch pper d forwrds o the write up. Scrtch pper: + + ( ) 6 + 6 6 We kow tht the lst lie is true becuse of our boudedess proof.
Officil proof: From our proof bove, 6. + + 6 + 6 ( 6) + + QED Puttig it ll together We show the sequece =, + = coverges. After lookig t few terms, we guess tht the sequece is icresig. Here is our three step pl. ) Fid the correct upper boud to use. ) Show tht this boud is upper boud. 3) Show tht the sequece is icresig. Scrtch pper: ) Fidig the correct upper boud: Lettig L = lim + = lim we get L = L (do you see why?) Solvig, L = 0 or L =. Sice the sequece strts out positive d icreses, we pick L =. ) Showig this upper boud works: We wt to show for ll by showig tht if, the + Strt from wht we wt to prove. + 4
3) Show the sequece is icresig. Strt gi from wht we wt to prove d work towrd wht we kow. + Now tht we ve lid ll the groud work, we c write the officil proof: Theorem: The sequece =, + = coverges. Proof: Clim ) for ll. Proof: = Assume 4 + Clim ) + Proof: From bove, we kow tht + +
From Clims ) d ), is icresig d bouded bove. By the Mootoic Sequece Theorem, the sequece coverges. QED Coclusio If you uderstd this lst exmple d your lgebr-fu is strog, you should be ble to hdle y recursively defied sequece.