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1 Koko Mrtoo FMIPA - ITB 7 Bris bilg rel Pegtur bilg rel dlm ideks terurut dimk bris. Bris bilg rel,,, ditulis { } =, tu disigkt { }. Secr forml, bris (tk higg) ii didefiisik sebgi fugsi deg derh sl himpu bilg sli. Ilustrsi Bris bilg rel yg poly, 4, 7, mempuyi rumus eksplisit suku ke- berbetuk = -, =,,. Dlm betuk rumus rekursif bris ii ditulis =, = -,. Bris koverge Bris { } diktk koverge ke L jik dpt dibut sebrg dekt ke L deg megmbil yg besr. Secr forml, bris { } koverge ke L, ditulis lim = L, tu Æ L jik Æ " e > $ NŒ ' N fi - L < e. Bris yg tidk koverge dimk diverge, mugki limity,, tu tidk d (oskilsi). Ilustrsi Bris { } deg kre ( ) Æ Æ = - ; { } 4,,,, koverge ke lim = lim - =. Perhtik situsi geometriy. ε ε = - ε ε y y= () = - 4 N N

2 8 Cotoh Buktik ( ) lim = lim - = deg defiisi limit bris. Æ Æ Bukti Ak dibuktik " e > $ N Œ ' N fi - - < e. Kre dik dicri bktk e e - - = < >, mk mbillh N bilg sli yg lebih besr dri e, mk N > e megkibtk - - = < e. Sift limit bris Utuk bris koverge { }, {b } d kostt k: () lim k = Æ k () lim k = k lim Æ Æ () lim ( ± b ) = lim ± lim b Æ Æ Æ (4) lim ( b ) = lim lim b (5) Æ Æ Æ lim Æ lim =, lim b lim π Æ b b Æ Æ Sift bris koverge Utuk bris { }, = f( ) ; jik lim f ( ) = L, mk lim f ( ) = L. Æ Prisip pit Utuk bris { },{ },{ } Æ L d cæ L, mk b Æ L. Utuk bris { } Æ b c, jik b c deg, jik Æ, mk Æ. Jik bris { } koverge, mk { } terbts. ({ } bris terbts jik $ M > ' M " Œ ) Jik bris { } mooto tk turu d terbts di ts, mk { } koverge. ({ } bris mooto tk turu jik " Œ ) Cotoh peggu prisip pit Buktik jik r <, mk r Æ. Bukti Kre r <, mk r >, kibty $ p> ' = p. Dri sii r diperoleh = = ( p) p> p" Œ, sehigg r <. Kre r r lim lim Æ Æ p = = (limit pegpity ), mk r Æ. Akibty berdsrk sift bris koverge diperoleh r Æ. p

3 9 Cotoh Buktik bris { } deg =! koverge ke. Bukti Kre { } bris positif, mk { } terbts di bwh oleh. Kre! ( )! = = =, mk " Œ, kibty { } bris mooto tk ik. Kre { } mooto tk ik d terbts di bwh oleh, mk { } koverge ke. (sift bris koverge) Cr li Kre < ( - )!, > 6 (buktik deg iduksi mtemtik), mk ( -)!!! < = < =. Kre limit pegpity, mk Æ. Deret bilg rel Dri bris { } butlh bris {s } deg s=, s=, s=,, s=. Bris {s } dimk deret bilg rel d ditulis =. Suku ke- dri bris {s } dimk jumlh prsil deret. Dri defiisi ii lgsug diperoleh = s - s, =,,, Deret koverge Deret = diktk koverge (puy jumlh) jik bris {s } koverge d diverge jik {s } diverge. Derety: Cotoh Selidiki kekoverge deret = ( ) 6 Jumlh prsil deret ii dlh = deg. = ( ) ( ) s k= k ( ) ( ) ( ) = ( ) Kre lim s lim ( ) = -. kk ( ) k k = = - = = -. = - =, mk deret ii koverge d jumlh Æ Æ = derety, ditulis ( ) =

4 4 Derety: Cotoh Selidiki kekoverge deret = 4. = = deg =. (deret hrmoik) Jumlh prsil deret ii dpt ditulis dlm betuk s= = s = = = 4= 4= ( 4) > s > / ( 4) ( ) s8 = 8= > Dri sii diperoleh Kre s ( ) Derety: Æ Æ > / > / s =. (buktik deg iduksi!) lim lim =, mk deret ii diverge. Cotoh Selidiki kekoverge deret () = - = - - deg () = -. = () -. { Jumlh prsil derety:, s bilg gjil = = {,,,, }., bilg gep Kre {s } tidk mempuyi limit (oskilsi), mk deret ii diverge. Sift deret koverge Jik deret koverge, mk lim. = = Æ Bukti Mislk jumlh prsil deret ii dlh s. Kre derety koverge, mk $ sœ ' lim s = s. Akibty Æ - - Æ Æ Æ Æ lim = lim ( s - s ) = lim s - lim s = s- s =. Ilustrsi Deret diverge kre = 5 - Æ 5 (kotrposisi sift deret koverge) lim - = π.

5 4 Ctt Keblik sift deret koverge tidk ber lgi. Cotoh peygkly dlh lim Æ = tetpi deret = diverge. Deret Geometri Betuk umum: - r r r = k- - k = - =. ( ) Jumlh prsil: s r r r r -r = = =, r π. s = r r r - rs = r r r r ( -r ) -r (- r) s = ( -r ) fi s =, rπ Jik r <, mk r Æ (hlm, prisip pit), kibty - ( -r ) s = r = lim s = lim = =. = Æ Æ -r -r -r -r Ctt Dri feome = -, = -, = -, diperoleh S = = - deg lim S =, sehigg deret geometri = 4 Æ = koverge ke ; Ilustrsi = - = - = ( ) =. = = = = = = 6. - (-) = - - = = = Ilustrsi = = ( ) Ilustrsi Jik <, mk Ctt Deret = = d =. - yg koverge ke S ditulis () - =. = = = S. Sift lier deret tk-higg () Jik c, mk = d c = bersm-sm koverge tu diverge. () Jik = S d, = b = T mk ( ). = ± b = S ± T =

6 4 Uji jumlh terbts Deret deret =, " Œ koverge s = k = k terbts di ts. Uji itegrl Utuk fugsi f yg kotiu, berili positif, d tk ik pd [, ) deg = f() berlku = koverge itegrl tk-wjr Ú f () dkoverge. y y = f () y y = f () Uji bdig bis = = Jik b " Nd b koverge, mk = b = koverge. Jik b " Nd diverge, mk diverge. Uji bdig limit Mislk, b>, d lim = L. Æ b Jik < L<, mk = d b = bersm-sm koverge tu diverge. (keduy koverge tu keduy diverge) Jik L = d b koverge, mk koverge. = Uji bdig Utuk deret = jik L<, mk deret koverge. =, > " Œ d lim Æ = L ; jik L> tu lim =, mk deret diverge. Æ jik L =, mk uji kekoverge tidk memberik kesimpul.

7 4 Aek Rgm Vrisi Cotoh Kekoverge Deret Cotoh Selidiki kekoverge deret. = - Cr Deg uji bdig bis, dri " Œ diperoleh -. - d = - Akibty -, sehigg - =. Kre deret diverge, mk deret jug diverge. = Cr Deg uji bdig limit, bdigk Æ b Æ b Æ - = - - = deg b =. Kre lim = lim = lim = < d deret diverge, mk deret jug diverge. d = Cr Deg uji itegrl, tujukk Ú diverge (kerjk ri- - ci prosesy!). Akibty diverge. = - Cotoh Selidiki kekoverge deret. = ( ) Cr Deg uji bdig bis, dri < " Œ diperoleh <, sehigg < ( ). Kre ( ) ( ) koverge (deret geometri deg rsio r = ), mk deret = = koverge. ( ) Cr Deg uji bdig limit, bdigk ( ) b = =. Kre deret ( ) = ( ) = deg ( b b ) = ( ) lim = lim = lim = < d Æ Æ Æ koverge, mk deret koverge.

8 44 Cotoh Selidiki kekoverge deret () =, (b). = l () Deg uji itegrl, kre b ( ) ( ) d b -/ -/ d bæ bæ bæ koverge. = Ú Ú = lim = lim - = lim - = (koverge), mk deret (b) Deg uji itegrl, kre d b d(l ) l bæ l bæ bæ diverge. = l Ú Ú b ( ) ( b ) = lim = lim l(l ) = lim l(l ) - l(l ) = (diverge), mk deret Hmpir jumlh deret Jumlh deret S= = dpt dihmpiri oleh jumlh prsil S= k d glty dlh E S S = - = k. k = b k= Deg kodisi fugsi f pd uji itegrl diperoleh E < Ú f() d. Cotoh () Jik jumlh deret koverge dihmpiri oleh = suku pertm, tetuk sutu bts utuk glty. (b) Tetuk gr glt dri jumlh deret S d jumlh prsil S plig besr,5. () Utuk deret ii pilihlh / f () = / yg berili positif, mooto turu, d kotiu pd [, ). Kre E ( - ) d b Ú = < = lim = =, k = / / / k bæ mk sutu bts utuk glty dlh,. (b) Ak dicri sehigg E = S- S <,5. Kre E ( ) b d - = / < / = lim k= k / = bæ,5 < Ú. mk E = S- S <,5 dipeuhi bilm. Dri sii diperoleh > 4, sehigg > 6.

9 45! Cotoh Tujukk deret koverge d hituglh =! lim Æ! Guk uji bdig utuk deret suku positif. Utuk deret ii ( )! ( )! Æ Æ Æ Æ lim e ( ) ( ). = d lim = lim = lim = lim = = <. Kre lim Æ <, mk deret deret koverge diperoleh! Æ! = Æ koverge. Berdsrk sift lim =. Cotoh Selidiki kekoverge deret. ( )! =!! Guk uji bdig utuk deret suku positif. Utuk deret ii ( )!!! ( )( ) ( )!!! = d = = = = >. lim lim lim lim ( )!( )! ( )! 4 Æ Æ Æ Æ ( ) Kre ( )! lim >, mk deret Æ diverge. =!! Deret gti td Betuk umumy dlh () 4, = - = - - > " Œ, suku-suku deret gti td berselg-selig positif d egtif. Ilustrsi () = - = ( ) - = - - = - = = dlh deret gti td koverge. (deret geometri deg rsio /) ( ) 4 = - = - - dlh deret gti td diverge. ( ) = - = - - dlh deret gti td diverge.

10 46 Uji kekoverge deret gti td Jik bris { } semu sukuy positif, mooto turu, d lim =, mk Æ () = - koverge. Ilustrsi Deret () - = - - koverge kre = 4 = > " Œ, { } mooto turu, d lim = lim =. Æ Æ 4 s s 4 s s s Tksir deret gti td Jik deret () s = - = memeuhi kodisi di ts d () s= - -, mk s - s. Ilustrsi Deret () = - - koverge ke s = d jumlh 8 suku pertmy dlh s 8 =, Tksir jumlhy memeuhi s- s =,646 < = =,965. Uji kekoverge deg ili mutlk Jik deret u koverge, mk deret u jug koverge. = Kekoverge mutlk d bersyrt Deret = u = diktk ko- verge mutlk jik u = koverge d koverge bersyrt jik u koverge tetpi deret u diverge. = Ilustrsi = Deret () - = - - koverge mutlk kre = = - = = = -. (deret ili mutlky koverge) Deret () = - = = deret ii koverge tetpi deret koverge bersyrt kre = diverge.

11 47 Ilustrsi Deret 6 si ( -) p = si 6 ( -) p = si 6 ( -) p d deret = si 6 ( -) p koverge. = = koverge kre deret ili mutlky koverge. Kre koverge (uji itegrl), mk deret Uji bdig mutlk Utuk deret, = π d lim = L ; Æ jik L<, mk deret koverge. jik L>, mk deret diverge. jik L =, mk uji kekoverge tidk memberik kesimpul. Pegtur kembli suku deret Suku-suku deret koverge mutlk dpt ditur kembli tp berpegruh pd kekoverge tu jumlh derety. Ilustrsi Deret () = - koverge mutlk berdsrk uji b! dig mutlk kre Æ Æ Æ Æ! ( )! lim = lim = lim = lim = <. Deret pgkt Betuk umum deret pgkt yg berpust di dlh = d yg berpust di dlh = = Ctt Dlm otsi ii Ilustrsi Deret geometri ( - ) = ( - ) ( - ) = wlupu =. = = dlh sutu - deret pgkt yg koverge ke s () = utuk < (tu - < < )

12 48 Himpu kekoverge deret pgkt Himpu ii terdiri dri semu di m sutu deret pgkt koverge d di lury diverge. Teorem Himpu kekoverge = sellu berbetuk: Titik = (selg [,]), jri-jri kekovergey. Selg ( R,R) (tu ( R,R], [ R,R), [ R,R]), jri-jri kekovergey R. Seluruh gris rel (selg (, )), jri-jri kekovergey. Teorem Deret pgkt = koverge mutlk pd iterior (selg buk) dri selg kekovergey. Ilustrsi Utuk deret!, uji bdig deg! = = memberik ( )! {, = L= lim = lim lim ( ). = = Kre L >, Æ Æ! Æ, π mk deret hy koverge di =., uji bdig deg =!! ( )! Æ Æ Æ Æ Ilustrsi Utuk deret! = memberik L= lim = lim = lim = lim =. Kre L <, mk deret koverge" Œ d selg kekovergey (, ). Ilustrsi Utuk deret memberik = ( ), uji bdig deg ( ) Æ Æ Æ ( ) Æ = ( ) L = lim = lim = lim = lim =. Akibty deret koverge jik L = < d diverge jik L = >, sehigg deret koverge jik - < < d diverge jik > tu <-. (-) Di titik bts =- diperoleh deret yg koverge. Di titik = ( ) bts = diperoleh deret yg diverge. Jdi selg kekoverge deret pgkt ii dlh - <. = ( )

13 49 ( ) Cotoh Tetuk selg kekoverge deret pgkt. Guk uji kekoverge mutlk deg ( ) = =, diperoleh lim lim ( ) lim ( ) ( ) lim Æ Æ Æ Æ L = = = = =. Akibty deret koverge jik L = < d diverge jik L = >, sehigg deret koverge jik - 5< < d diverge jik > tu <- 5. (-) Di titik bts =- 5 diperoleh deret yg koverge. Di titik bts = diperoleh deret yg diverge. Jdi selg kekoverge = = deret pgkt ii dlh - 5 <. Opersi pd deret pgkt Turu d itegrl suku demi suku deret pgkt di iterior selg kekovergey. (iterior dlh selg buk terbesr dri selg kekovergey). = d - = d = = = Jik s () = = pd selg I, mk () ( ) s = = =, iterior I Ú s () d = Ú d = =, iterior I Ilustrsi Dri deret pgkt (- ), - = - < < diperoleh = 4,- < < deg cr meuruk suku demi suku deret semul. Ilustrsi Dri deret pgkt 4, = < < diperoleh l ( ) = - -,- < < deg cr megitegrlk suku demi suku deret semul.

14 5 Teorem Abel (Kekoverge deret pgkt di titik ujug selg) Jik s() =, (, ), Œ -R R s kotiu di R d R, sert deret koverge utuk = R d = R, mk di titik ujug selg berlku = R = sr ( ) d ( - R) = s( -R). = = Ilustrsi Dri ilustrsi terkhir kit mempuyi deret pgkt 4 l ( ) = - -,- < <. Kre fugsi y = l ( ) kotiu di = d deret di rus k koverge utuk =, mk 4 l = - -. Cotoh Tetuk deret pgkt utuk t - d sutu deret utuk π. Dri deret pgkt diperoleh, - = - < < gtilh deg -, 4 6 dri deret ii meghsilk = - -,- < <. Itegrlk suku demi suku Berdsrk teorem Abel, kre rus k koverge utuk =, mk t = - -,- < <. - y= t kotiu di = d deret di - p t = = - -, sehigg sutu deret utuk π dlh 4 ( ) Cotoh Tujukk 5 ( 5 ) p = - -. l - =,- < d tetuk sutu deret utuk l. 4 = - 4 fi it fi it 5 = - - l( ) = - -, - < < -l(- ) =, - < < Ambil ( 5 ) =, diperoleh l l - ( 4 5 ) 5 l, - = - < < = =.

15 5 Opersi ljbr pd deret pgkt Du deret pgkt yg koverge dpt dijumlhk, dikurgk, d diklik suku demi sukuy seperti pd sukubyk. Du deret pgkt yg koverge jug dpt dibgi seperti pembgi pjg pd sukubyk. Ilustrsi Tetuk jumlh, selisih, hsilkli, d hsilbgi deret pgkt, = < < d, - = - < <. Jumlh: Dri = (- - ) ( ) diperoleh - = ( ), yg meghsilk Selisih: Dri =. - = (- - ) -( ) diperoleh 5 - =- ( ), yg meghsilk Hsilkli: Dri - diperoleh rus ky dlh 4 =. 4 4 = (- - - ) ( ) , 4 yg setelh disederhk sm deg. Dlm ksus - ii jug dihsilk 4 =. 4 4 Hsilbgi: Dri ( ) ( ) - = deg pembgi pjg diperoleh rus ky dlh dst. Dikerjk tp proses pembgi pjg, kit mempuyi ( ) = = = - = -- - = - -

16 5 Deret Mcluri Perhtik proses meetuk koefisie deret pgkt f () = diytk dlm turu dri fugsi f pd selg = ( R,R) deg R jri-jri kekoverge deret. f() = = fi = f() = 4 4 f () = 4 - fi = f () f () = 4 ( - ) - fi = f () f () = 4 4 ( -)( - ) - fi = f ()!... () () f () =!!( ) fi = f ()! () f () Akibty kit mempuyi f () =,- R< < R, yg dikel sebgi deret Mcluri (di sekitr ) yg koverge ke =! f. Deret Tylor Jik f () = ( ), -c - R< - c< R (c- R< < c R), = () f () c mk deg proses yg sm, f () = ( -c), c- R< < c R. =! Deret ii dikel sebgi deret Tylor di sekitr c yg koverge ke f. Cotoh Tetuk deret Mcluri d deret Tylor di sekitr c utuk fugsi f () = e. Kre dlh () () =! f = e deg e f () () =, mk deret Mcluri utuk fugsi ii =. Kre deret pgkty koverge,. = e = = Œ c! 6 " Œ, mk Kre f () () c = e, mk deret Tylor dri f () = e di sekitr dlh c e! ( ) e = -c, Œ. = Deret terkhir dpt jug diperoleh deg cr c -c c ec = =. e = e e = e ( - c ) = ( - c ), Œ!!

17 5 Ilustrsi Tetuk deret Mcluri utuk f () = si d g () = cos. Dri f () = cos= si( p ), f () =- si= si( ) p () f () = si( p ), sehigg f () si p,,,,, () diperoleh = =. Jdi deret Mcluri utuk fugsi f () = si dlh 5 7 =! 5! 7! ( )! f () = si = - - = ( - ), Œ. Alog: g () cos( p ) () = d () deret Mcluri utuk fugsi g () = cosdlh g () = cos p, =,,,,. Jdi 4 6 =! 4! 6! ( )! g () = cos = - - = ( - ), Œ. Rumus Tylor deg suku sisy Jik fugsi f mempuyi turu smpi tigkt-() pd selg buk I yg memut c, mk " Œ I, () () ()!! f c f c ( ) f ( ) ( )! c f () c () f () c!! f ( ) = f( c) f ( c)( - c) ( - c) ( - c) R ( ), deg suku sis R () = (- ), ξ di tr d c. Di sii P () = f() c f ()( c - c) ( - c) ( -c) dikel sebgi sukubyk Tylor d () R suku sis Tylor. Teorem Tylor Jik fugsi f mempuyi turu di semu tigkt pd selg ( c- r, c r), mk deret Tylor () f () c =! ( ) f ( ) ( )! ( -c) dlh uri fugsi f lim R ( ) =, deg R () = ( -c), Œ( c- r, c r). Æ Deret Biomil Utuk yg memeuhi - < < d " pœ berlku p Êpˆ Êpˆ Êpˆ Êpˆ p( p-)( p-) ( p- ) ( ) = Á, Á = Á Ë Ë Ë Á Ë! Ilustrsi Deg rumus deret biomil, / ( )( ) (-) 5 ( -)! =! ( ) = - =, <.

18 54 Hmpir fugsi deg sukubyk Tylor Jik fugsi f mempuyi turu smpi tigkt-() pd selg buk I yg memut c, mk " Œ I, f () = P() R (), deg d f () c () f () c!! P () = f() c f ()( c - c) ( - c) ( -c) ( ) f ( ) ( )! R () = ( -c), Œ( c- r, c r). Utuk = kit mempuyi f () ª P() = f() c f ()( c -c), dikel sebgi hmpir deg gris siggug. Utuk = hmpir deg fugsi kudrt, d seterusy. Utuk R () yg terbts dpt dihitug bts keteliti hmpiry d besry gr hmpiry memeuhi bts glt yg diberik. Cotoh Hituglh hmpir utuk e deg glt plig sedikit Uri Mcluri dri!!! e d suku sisy dlh e = R ( ), c e ( )! Utuk meghitug e mbillh =, mk diperoleh e!!! = R (), R Adik e <, mk Crilh sehigg -6. R () =, c di tr d. c e ( )! () =, c di tr d. c < c< fi < e < e < fi ec R () ( )! ( )! ( )! -6 6 < R ( )! () <. Kre - ( )! 9!! < = <. < <, mbillh ( ) =, sehigg = 9. Jdi hmpir utuk e deg glt plig 6 sedikit - dlh e = =,788.!! 9! Cotoh Hituglh hmpir cos 6 deg sukubyk Tylor derjt du besert sutu bts utuk glt hmpiry. p p 4 = 6 = p p 9 p p sic R R( p 9 p ) (! 9 p ) ( 6 9 p ) Dri ( ) ( ) cos = R ( ) mbillh mk diperoleh = - ( ) - ( ) R ª deg bts glt cos6 ( ), R ( ), ( ) = = < ª,7.,

19 SOAL LATIHAN MA KALKULUS A Pokok Bhs: Deret tk Higg 55 Sol uji kosep deg ber slh, berik rgumetsi ts jwb Ad. No. Peryt Jwb. Jik b" Œ d bris { b } koverge, mk bris { } koverge. B S. Jik bris { } koverge, mk bris { / } koverge ke. B S. Jik bris { } koverge deg lim Æ Æ Æ = L, mk lim 4 = L. B S Æ 4. Jik lim ( - ) =, mk lim d d iliy higg. B S 5. Jik deret Σ diverge, mk bris jumlh prsil dri derety tk terbts. B S 6. = ( l ) dlh sutu deret yg koverge. B S 7. ( ) ( ) ( ) 8. ( - = ) <. B S dlh sutu deret yg koverge. B S 9. Jik deret ( - ) koverge di =,, mk deret koverge di = 7. B S = = = () fd =. Jik f () = d deret koverge di =,5, mk Ú. B S Selidiki kekoverge bris berikut d tetuk limity bil koverge 9 cos ( p ) l. =. 9 =. = 5. ( ) = - 6. si = 7. p = 8. si = = 4. ( ) / Tetuk jumlh prsil deret berikut d jumlhy bil koverge 9. ( ( ) ( - ) ). e ( ) = r ( - r ), < r <. ˆ = = p Ê - = Ë( -) l = 4. l ( - = ) Selidiki kekoverge deret berikut deg uji kekoverge deret suku positif = = (4 ) 7/6 7. e - e 8. = = e 9. = (l ). si. = =. 8!. 4. =! = 5. = 6. =! = =!

20 Tujukk deret berikut koverge ke S d tetuk hmpir S S ( ) - 4. ( ) = - 4. = l ( ) = - ( ) l 56 Selidiki pkh deret berikut koverge mutlk, koverge bersyrt, tu diverge. 44. () = () () = ( ) - = () () - = 5. ( ) = - 5. = Tetuk himpu kekoverge setip deret pgkt berikut. () = - l si () = - ( ) - () - = = ( -)! () = - ( ) 54. () = - 55.! ! 5! 7! - Tetuk deret pgkt dri fugsi berikut besert jri-jri kekovergey. 6. f () = (- ) 6. f () = - 6. f () = Ú f () = l( t) dt Tetuk deret Mcluri smpi 5 d deret Tylor smpi ( ) dri fugsi f (). 64. f () = t 65. f () = e si 66. f () = (cos)l( ) 67. f () = e si cos f () = 69. f () = 4 7. f () =, = 7. f () = e, = Sol Aek Rgm 7. Pd deret () = d (b) = 4, tetuk gr E= S- S<,. / - 7. Tetuk sukubyk Mcluri derjt utuk ( ) d bts glt R () jik,5.!! 74. Deg megguk deret = buktik lim =. Æ Kuci Jwb. S. B. B 4. S 5. S 6. B 7. B 8. S 9. B. B... diverge 4. e 5. diverge e π diverge.. 4. l 5. diverge 6. koverge 6 p( p -e) 7. koverge 8. koverge 9. diverge. diverge. diverge. koverge. koverge 4. diverge 5. koverge 6. koverge 7. koverge 8. koverge 9. koverge 4. diverge 4. S- S9,65 4. S- S9,47 4. S- S9, 44. k. bersyrt 45. k. mutlk 46. diverge 47. k. bersyrt 48. k. mutlk 49. k. bersyrt 5. diverge 5. k. mutlk < 56. < < < 59. < < 6. 6 ; ; ; ; ( - ) 4( - ) ( - ) e e e e( - ) ( - ) ( - ) 7. () > 5, (b) > d R ( ),5

21