Lec 23
rc/cpd 1 Set TylrsThm_ tccli k c k cepmcc YY} ẋc cth R clhcc+hpecctwtyty?_ht + E where 3 ete c d c+h 31 c + O cth c 1 Reccth +@ligm ht Schlidmilch 11847 Rc h Ttmh m k+h m ti e 1 d the cse m Lgrge m Cse m RcCctHMyYgthhI@jhtc3yeg Cuchy s m c llh t O c Tylr 4 1 47 C et *C 2 34 72 @Tp ED C! C1 series II LI @D! c IR im/e3/ceyy*ieyht/l*eitt Fr 141 / IT 1 R CH/sh11hlt S llhh e1 hkk
imil l 1 P ktl eig equti let AIR HX XEC c l H C c + 1 it 2 6 k 1 m c D k+d HX k lklc l D Tylr series 2 d D c ko ktl k X Deie 21 1 F kti Te kzl Tylr ± series HX& Tylr ply R h + h E k d k k
Q t des }P Thm? R dieretile re ucti cverges t k }P d g there uit ets tht l tht p Here gl Seri Crllry Let }? Th ± sii cv At TEuhcu uimly ui cverges d t ucc E Fzuc U Uc FI Xm cv ucti 5_ X uic F IR I R / i O µc ui ui NOT Cuchy uit perspective usig 11 llu rm suppse c SH cc Zuic I d i cvergece The O itpm C p O cverges uit 1 11 11 Als c z NOT O dieretile e R 2 g cv 3 seq c ll ld l dieretile ui the ktii lzi 1+7 h H222 NX 2 th Here 9l eg ce lc d cv ucti }P tht suppse 1 cl
By ve crllry the t 9l De ; p C I 1 d mesh i Xk lpl let OXK M k lpl vlued i rt k k Riem 12 mesh p tr ctk % tptfei?kgp P ; eve with Let prtiti icresig su itervls itervl su rt r iiitesiml k A lttpp sy sequece prtit it k1 ± mid Yi Mk ctiiiij Fr t ech Ps Mr spc cj eiiiiiij hve SC p tie Emie i k1 iite Y s itegvk d ets Riem hermit + dete tptr tplpjl cj y tke ech SC suppse } leks sum S Typiclly the I ucti prtiti i iiitely reied mes 12 rel k Xkt uded } e Chse y hece tk }
time itegrte ver FE te mkk y EE mi temyir s miiyii p± I ctiuus He Tke sice ye& y1 uit ui ct itegrk Et the cy i set cmpct chse ct ct tht IS ucti ctiuus ly Ky E@iemiiyieIEiel s 1 Crllry Give I seg whe Pj prtiti j lpjl iriiij Ye Ms jhulpj 1 tht j y hve ir ye ki j k1 p± I e itegrle mtic suppse with prtiti eve u WLOG * ud whe j B ve the Tke Ecmiiimieli s t Thm t Crllry zcmii mid Yi He k1 summrize T kl mie S s ly lc su s the icresig itergrk itervls / Mk Xk mk k 1 Xk lkt 5
mkk @ Suppse pits Thi I l Cii eg T g Cu cii By i Fudmetl Thm_ Let Thm Clculus eitegrk Fc t F tti Ii p the ctiuus ceir i the M s the Ml ; $ e ll Il iths Fr Ltdt cstt E M itegrk c R c d F AEXEB put ucti ctiuus eep itegrk ids the s d IFIEM CECA Tke M itegrk i itegrk ll dctiuus my prticulr I iii iitely ly h q creitegrk eg ig Ci with itegrte itegrte th re c s itegrte g hl Let Tim Suppse uded Thm_ l u D ut k dieretile t Furthermre d X i Fk c ct
t Ft i c FHDI ct Fy PI Sice itegrte i Set s M lctsl te uded Give e ys hve / Fcy Ltdt/ FH / utdt gpdt E M y i ly XI ME hve IFCY ME E Th prves tht F ct ctully uit Suppse ct Wt t shw F e Ci Fl l s udt ctdt X X esuplittcl 11 dt 11? Yddt_ ttl H E I 8 whe c Xts sup E Ict c ttl FlEF L i / 5 ttihmdt_ g spllt ttl similrly lessfgp e µ FK i t c } Ft
Flk i d Fl Fl Fu Fl Thi I itegrk i there dieretile ucti F i F them c d Fl p p t PI F E ki Mk sice itegrk c id e prtit mk k E t O ech Xkt XD kl i the MVT Flk Nte FHDOXK I ltrdk where tke k d ct 2 Xr Fcr smltk Mk Fl l 5 d e Fl IF d E Mr Xk mk k E Sice ritrrily smll hve Ft hd B * Itegrti sustituti * Itegrti prts