59_20111121mon_calc2_94159_f11

59_20111121mon_calc2_94159_f11 - FALL 20 u CALCIL Mo.1 2...

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Unformatted text preview: FALL 20 u CALCIL __ Mo .1! 2!. MAC 23%(2-159 22/2 SWQWE. @ £33 Flhah'amg §¢M : Sguenoes 19.240), E] Defis. m A Seqyence {an} (5 bounded from above {3? there. 87((5‘5 a, number M gap/4 that“ 3 CL g aha-M fora" n (is) El all: aném :Far‘ n=l,2,s,... O w? cut. aha/w We? 00.) mg __ e? ~———N ——m,w ZLIJ E] Any number) M1 which gahsjcc‘cs ineqyalh‘g (s?) (9 cafleok an ugger bound fir {63“} ‘ ——_—___—_—‘AV ————————————————M’ Luis. (pew Ukper Bound E] Dlécussion- Suppose {09113 is a seiuevtce which is bounded éllvcsveJ and suPpose M is an ”H’er bou 4. 55? {an}. Then :4 show! be clear {MP MH IS also an upper bound for {an}, as are M+ i1) M+fijand MN? For exEMPl'f. 'iQn} W/ an: “"7!“ is bounded above) smce (-4? 4 QJVVIéZJ’ $6 2 (s an um“ bound. BM’ so qr: 3J 6,“), and lo 071- er But More Iv‘npb H1 the followmq are 549° 0 yet bounds fir {1‘35} 3) P (.5) l.25) mzs) 3.05.25J and so cm. film} [\I‘ So: Hm. mfiemw‘cim @341». nm‘urall asks {he @9957me “Is there a ‘SMalles’c’ UFPG" bound 22v the sea}; me? I [8 M is am U rboand IQ ‘Lf’. n j filfmber smaller? an M is aflp‘éefimflfia for $64.1} than M (‘5' Called fie Lgé Umr ELM and {cm lab {an} =M. le .IE ‘ Mcflecreasin Se uence T14?” (p. 740) A,“ “M ecoec‘re‘lsmiys eclyevzce cam/erg es «f and em: t__€ H" t5 bouhd 5‘ {hum abOVe. [a 1X3 “°""°\ecr€35i"3 SGiUehce converjes) {idem i-(- conver3es +0 1 (e334. Upper bound. No Prom; given here. (2:! Nm‘afi‘on: (El ‘\ .4: and (MN w” : PF? , <=> This is flat s+rbngesf [osl'cal conned-(oh. 40<=> qy means 07:}in /\ (1:) P3 “IF f9)’tl/1€K ‘Lj and (F (L) "then 40. ” “ (9&er upper bound " : lab) I-q.b.J L.u.BJ LUB (a {ad is Nondecreaswtg (MeaMS- a! £612 5 aséa4é,,, fan—an.“ —oao an - M ,g cm UPpCf EMM- P I( I: n I! ML. I? #16 smaUegf' or 30 U fer bounds Mamas - g“ 2 L" ngihik Smes. g E 7465.[1You are gdchq gamewuing up? WE Ewamwe {ah}: {2'7} (”we (50“ jarsf 6+erm5. ‘ j } _L (, __ I r 2)4/3)"z;§§,z%— 6312030“) £32442.“ ‘1“; [g a. hohMcreasm 56 em: ,3 H— has a, if“; (firedesJ' ‘ 4' lower bout/13H), 2—,1 _y? 0 1 {:7 I Wall/1+ +0 4"} "('0 “dd: UP ” fie 4am; 5F fie Seiuence' [:3] Hernia/9 (AN/10$ ML [Mk9] like (In general) "2:; ah gang 0-. +42 +43 +CL4 4"... 4 Bed' vol/m!" does —/fi¢symean7 {E 1.9% {an} be CL seq’uemce, m We dept-M 616 “-11 yarh‘al Sums ENNNN SW 7-“ a‘+a‘z+m+qn 52$ ,4: vvvvvvv mmmmmm mmccccmc: <<<<<< SOQU SHEETS FlLLER 5 ECU _ A _ c , CA M, 52,1?” ZJ 3‘6 H. E ., leadsa‘u h a1] saw w/ {ma—9 {9"} my sn=Zaz H k=l I< Am we define Zah = hm 24K- nr>do (4:! m. a.) amt“ a,+«z+a3j a,+4z'+4g+a4j SI) 52, 5?) 5.4-)l-q 1: 361% L ) fig (97 d9?“ Zaw = L n:( {E W (CW3 14mm we Gecmd-n‘c gamma) {0“} Gewwgg, 0': a, 42:4.» Q3 =arr :1 arz 0. —ar3 r: common I‘d-10- 4- yaw“; ~_’/‘ ODENDG“ ...
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