terms of discrete sequences can only take values that are discrete as opposed to

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1 Diol Bgyoko () OWER SERIES Diitio Sris lik ( ) r th sm o th trms o discrt sqc. Th trms o discrt sqcs c oly tk vls tht r discrt s opposd to cotios, i.., trms tht r sch tht th mric vls o two cosctivs os r sprtd by mbrs tht cot b vls o y othr trms o th sqc. For istc, th mric vls o trms, or,,,, c oly b,,,... btw d thr is o trm o th sqc. So, mbrs sch s.9,.8,.7,.6, c b vls o trms o this sqc. Th sqc o trms:, r, r... or r itgr is lso discrt i r is giv id vl. I r is llowd to tk y vl, th this sqc, lik th ollowig o, bcoms cotios sqc. It is sqc o ctios s opposd o sqc o mbrs. Sppos o tks th sqcs p, p, p... whr is ot cssrily itgr. Th, dpdig o th vl o, o c id trm o th sqc to b ql to y giv mbr. This sqc c b clld cotios sqc s opposd to discrt o. Lt ( )... b th sm o cotios sqc. Sic o c cosidr y o th trms or s ctio

2 Diol Bgyoko () o, th sris ( ) is sris o ctios. [i.., th trms o th sris r ctios dpdig o vribl which is cotios] owr Sris By diitio powr sris is o th orm: ( ( ))... whr th pots o r positiv itgrs or zro. Not wll tht som o th coicits my b zro d tht som my b compl mbrs. S D ( )... is powr sris S E 6 7 ( ) 8... is lso powr sris; so is S F ( )... Not tht th powr sris S F () is ot gomtric!!! sris! costt. This is so bcs th coicits lso chg. Covrgc o owr Sris Lt S p () b powr sris. W wt to kow i it covrgs. By th rtio tst, w hv I lim <, th sris covrgs. So w mst hv <. Empl. Lt or > Th, So th sris covrgs i <. Th sris covrgs or ll sch tht <. Th st o rl mbr C {, < }, or which th sris covrgs, is clld th domi [or i this cs th circl] o covrgc o th sris.

3 Diol Bgyoko () Empl. Lt... Th, th sris is simply r, circl o covrgc: {, < }... This is gomtric sris. [i.., ths, i <, th sris covrgs. Agi, i this cs, w hv th C. Empl. Lt powr sris b whr.!! ( )!! So, th powr sris! ( )!! NIFORM CONVERGENCE Lt S ( ) S ( ) i Diitio: i i. For y iit, Lim <. covrgs or y iit. So, C {, iit} ], [. b iiit sris o ctio d lt S () th prtil sm:.... A sris (grlly o ctios) is sid to b iormly covrgt i ε smll d positiv, N, sch tht or > N, o hs S S < ε I ordiry lgg, sris is iormly covrgt i, giv iiitsimlly smll mbrε, thr ists itgr N (lrg) sch tht, or > N, S ( ) is iiitsimlly clos to S, th sm o th sris. Th bov sttmt ssms tht th sris covrgs (S ists d is iit). Not tht iorm covrgc is did or ll sris o ctios [I m ot syig tht thy r ll iormly covrgd]; it is ot or powr sris oly. Howvr, th ky s o iorm covrgc is grlly with powr sris. CT (iorm Covrgc Thorm):

4 Diol Bgyoko () A powr sris is iormly covrgt i its circl o powr sris is covrgt, th, it is iormly covrgt. covrgc. [ ltrtivly : I I othr words, w r syig tht thr r sris o ctios which c b covrgt withot big iormly covrgt; bt, or powr sris covrgc implis iorm covrgc. SERIES EXANSION OF A FNCTION Giv ctio () or (z), c w id sris whos sm is (). Giv (), is thr ( ()) sch tht: ( ) ( ). I th () r moomils, i.., th w tlk bot i powr sris psio o (); i th () r, cos or si( ), w tlk bot Forir psios. Not vry wll tht i ( ) ( or z) ) ( ( z), th i most prcticl oprtios w will pproimt () or (z) with prtil sms S )... ( ) or S z) ( z) ( z)... ( ). ( Thr lis ky importc o sris. ( z How lrg shold b so tht S () is vry good stimt (pproimtio) o () will grlly dpd o th vl o. Not lso tht i th stdy o sris psio o ctios, o hs to mk sr tht th sm S o th sris is ql to th ctio. (Crlss is dd bot this poit s show blow!) Empl W shll show ltr tht

5 Diol Bgyoko () ( )... ( )... Not tht or > th sms o th right r ot ql to th ctios! Thror, ths two psios hv mig oly or <. For istc, or, ( ), bt th sm o th right, or, is iiit! THE NIQENESS THEOREM (A Etrmly sl thorm) Thorm: Th powr sris psio o ctio, i it ists, is iq. This trmly powrl thorm simply sys tht yo c gt th powr sris psio o ctio by y mthod yo choos. Th rslt yo gt is th sm s y othr mthod will giv. My stdts lwys srvy vry qickly th mros mthods (Tylor, McLri, dirtitio, itgrtio, biomil ) d pick th shortst or gttig th powr sris psio o giv ctio. Not som hv ild my ms i th pst, or ot doig so; I s o rso why som will ot il i th prst or tr lss, o cors, thy rliz tht srvyig ltrtivs is idispsbl procss i itllctl dvors. As sl, I m lookig orwrd to As rom my who r dvlopig itllctl rigor! [It is lik gttig i i 5 withot sig th polr orms o i d I s w shll s] 5

6 Diol Bgyoko () OWER SERIES EXANSION METHODS rlimiris rcll th qstio: giv (), is thr sris ( ()) sch tht ( ) ( ) [or ( z) ( z) ]. Th swr is giv by th Tylor thorm i th sris i qstio is powr sris. [I th sris i qstio is o si d cosi ctios, th Dirichlt thorm provids th swr. Tylor Thorm (compl or rl ctios) Lt () b lytic (or holomorphic, ) isid circl C o rdis r, ctr t z., ' ( z ) ( z ) Th, t y poit z isid C, ( z) ( z ) ( z z ) ( z z )... This!! thorm hs hidd sid to it. Wht hpps i () (z ) dos ot ist? Z Hidd spct : It dos ist bcs i ctio is lytic t z, it hs drivtivs o i i i Z C ll ordrs t z which r lso lytic. This thorm pplis to ctios o rl vribls i s wll (s yor clcls book). Tylor thorm (rl ctios): i () hs drivtivs o ll ordrs o sgmt ( ) ( ) ctrd o, th, or ll isid th sgmt: ( ) ( ).! 6

7 Diol Bgyoko () Applictio: Fid th powr sris psio o rod. () ( ) ; ( ), ( ). sig th mthmticl idctio blow, w gt ( ) ) ( ) ( ) (!. Not tht!. Mthmticl Idctio: W hv vriid by dirct clcltio tht ( ), ( ), Lt s ssm tht ( ). From this, w hv to prov tht d ) d ( ) ( ) ( ) d d (, ths, ( ). So, or,, ( ). Th powr sris psio o rod is thror: ( ) ( ) ( )...!!! ( )! Mclri Mthod: Th Mclri psio is simply Tylor psio with [or z ] Applictio: Fid th Mclri psio o. ( ), ( ), by mthmticl idctio w show, s prvios mpl, tht or ll, ( ) so () d ( )! ( )!....!! Fid th Mclri Sris or si(): 7

8 Diol Bgyoko () () () () () ( ) cos, ( ), ( ) si ; () ; ( ) cos, () () Si() d () (5) (5) (5) () () (), ( ) cos, ()!Not tht () () [ ( ) si ] (6) (6) (6) () ( ) si, (). Not lso tht () () sig th mthmticl idctio, w show [yo do it!] ssmig () () ( ) ( ) ( ) ( ) cos, ( ) si, ( ) cos ( ) d ) si, tht ( ( ) ( ( ) ( ) ( ) ) ( ), ( ) ( )... Th bov will prov tht ( ) ( ) ( ) [si( )] ollows cycl or. So 5 si( )!! 5! 7! 7 si( )! 5 5! 7 7! ! ( ). ( )! si( ) ( ) ( )! Th Biomil Thorm (Biomil Epsio) d Applictio For ( ) rl d < ( )! ( )( )!... Fid th powr sris psio o bot. ( ) with. As th psio is bot zro, w limit vls to < ; th by th biomil thorm: ( ) ( )! ( )( )! ( )( )( )!..., ( ) ( )( ). ( )( ) ( )( )( ) 6. 8

9 Diol Bgyoko () So ( ) !! 6 W sd th ct tht,,... Similrly o ids tht!! A compct orm o th rsltig powr sris psio is s ollows: ( ) ( )..., c b pt i compct orm! oc w di th prodct π ottio: π, π i..., π i...! so ( ) π i d i ( ) i ( ) i i π i ( )( ).... So th biomil psio c b writt ( ) π c! c ( ) Not wll tht th biomil psio givs iit sris (iit mbr o trms) i positiv itgr. (Jst vriy this by d or yorsl ll it tks is workig ot mpl with sy or ) Also ot wll tht ( b) b( ) b ( ). b b So, i b th b b ) b to pd ( ) i <. b b <, ( ) ( with <. INTEGRATION OR DIFFERENTIATION METHODS b O c s th biomil thorm 9

10 Diol Bgyoko () Mthods ), ) d ) pply mostly to powr sris oly. Th itgrtio d dirtitio mthods pply to iormly covrgt sris o ctios, i grl. Rcll tht iorm covrgc, or powr sris, is implid by covrgc. So, lt th 5 7 McLri sris or si() b: si...; id th McLri sris or! 5! 7! cos. d si d cos! 5 5! !! so cos ( )!!! 6... or 6! cos ( ) W kow! d () cos ( ) ()! Giv ( ) ( ), id th McLri sris psio or l( ). d W kow tht [l( )] or tht l( ) d d d d... d So, l( ) ( ) OTHER METHODS

11 Diol Bgyoko () I told o o tht th bov mthods r th oly os. I did ot itimtd ithr tht som o tody s stdts will ot costrct othr mthods mor powrl d lgt th thos sd by thir tchrs. Hr is vry sl sbstittio procdr: Fid th McLri sris or Lt...th!! 6...! Sm qtio or z, z iy.gt z z z! z!... θ ( iθ ) ( iθ ) A pplictio o this lst rslt givs: i ( iθ )... or!! θ θ θ θ θ θ θ i... i θ... cosθ i siθ!! 6!! 5! 7! Th bov rcis provids proo o th Moivr-Elr rltio: iθ cos(θ) isi(θ) Not A ctio () is v i () (-) or ll vls o. Empls: cos,,... A ctio () is odd i, or ll vls o, () -(-). Empls o odd ctios icld si(),,, 5, tc. Not wll bov tht th powr sris psio o v ctio cotis oly v pots o th vribl. owr sris psios o odd ctios coti oly odd powrs o th vribls.