Emhases o Calculus Ininite Sequences an Series Page 1 Sequences (b) lim = L eists rovie that or any given ε > 0, there eists N N such that L < ε or all n > N ¹, YgM L íï böüÿªjöü, Éb n D bygíí I an { } is boune above I an { } is boune above I or an { } is boune, then { } converges. Let () be an arbitrary unction an eine := (n) or n N. Then, šn? ʪ A 0 0 orm C lim () = L lim = L orm í8 v «à L Hôital Rule JjZ lim () JbuJ]cjì,.? )w close orm, lÿ ávhô, O.øì?Dÿ iygd øìbêüìygí8 - (cq íygmñ L), nª* ]cì cqêþê n ví8 (øf, 1, J L H) Zª YgM Series (b) I n-sum s n := n a i, ì e a i s n. FJ a i = a 1 a a 3 Éu$[ý,.øìæÊ. bygíí n th -Term Test : I converges, then lim = 0 ( ulsì bygí.b K!) ø n YgD, lõ lim u Ñ 0, à = 0, n./àwf, Jn ÌŠ ½µ,øõ, J n=1 Yg, lim = 0; <¹: Ê n ' ív` -±B0 í cq() Ê [1, ), / / 0, J := (n), ¹U.uÀ ]Á, ÉbæÊø_ cbm U) () 0 or > M,, à integral test Võ: M 1 = YgD, YgD, 1 n=1 () = n=1 Ìáí,< M () 1 < Ê [1, ), / M () YgD, () YgD, M Coyright c 004 by Dr. Mengnien Wu : mwu@mail.tu.eu.tw
Emhases o Calculus Ininite Sequences an Series Page ], ø Ì b YgD, ; ÌâÜ} Ìáí (.MÖ, n= íáb (Ì Öá) eñ, M 1 n=1 íáb (M 1á) eñ ø ) b ( n, n b n, 0, b n 0 n) ög 5^z 1. Integral Test : := (n), () 0 an continuous on [, ). Then, n= converges () converges.. -series test : I > 1, then 1 n converges; n I 1, then 1 n iverges; n 3. Orinary Comarision Test : 0 b n or all n. Then, bn converges = converges, i.e. iverges = b n iverges. 4. Limit Comarision Test : Let r. <u, Ê n /' í M 5(, u b n b n í ri FJ r b n = r b n. 5. Ratio Test : let r. <u, Ê n /_' í M 5(, 1 u í ri FJ a M a M r a M r a M r 3 = a M r n. 1 6. Root Test : let r n. <u, Ê n /_' í M 5(, u r n FJ r n. Ratio Test Root Test u* Sb íiv«ní, @v.øüjnu (Uàvœ Ñ: 1 ª Ö (_b) 7 n í a) (ÉÝ b) 1. úlsb, "úyg Yg ¹: J b n Yg, b n ( ) Yg (ÄÑ 0 ), FJ n n ( ) n, Yg "ú.} n Yg 7 n êà í8. > b Y.Ygu'ñq í (â n th -term test). ª?u KYg, 6ª?u " úyg "úygíb, Ìuú n C n 7, ¹UL< ² áíßå 6.} àwygm OBb.ø. ½ à íb <, Bý KYgíb ÿ.?óz½ 3. øøbu Ñ "úyg, Éâø bög 5b,"úM ¹ª n=0 Power Series (4b) Ì () u ÑÖá, I 0 () := (), BbªJì n1 () := n() n(a) a, a, ¹ n () = n (a) n1 ()( a), a à øò.ië ó Î-, ÿ})ƒ () = 0 () = 0 (a) 1 ()( a) = 0 (a) 1 (a)( a) ()( a) = 0 (a) 1 (a)( a) (a)( a) 3 ()( a) 3 Coyright c 004 by Dr. Mengnien Wu : mwu@mail.tu.eu.tw
Emhases o Calculus Ininite Sequences an Series Page 3 FJÛÊÿ)½: n (a) <b ubó J n Ê a Tª, ;W L Hôital Rule, n1 (a) := lim n () n (a) n1() a a a a n() = n(a) Ä, () (a) J 0 () Ê a Tª, 1 (a) = (a); a a J 1 () Ê a Tª, (a) 1 () 1 (a) a a a () (a) (a)( a) a ( a) () (a) a (a) a () (a) a ( a) () = (a), Kwõÿu () Ê a Tu Ÿª a 1 énë, J () Ê a T n Ÿª, ªJ)ƒ n (a) = (n) (a) J () Ê a TÌ Ÿª, () $,ªŸA: () = (a) (a) 1! ( a) 1 (a) ( a) (n) (a) ( a) n U iê () í ower series (4b) in ( a) C Taylor eansion o () about a(a = 0 v ± Maclaurin series), Yg UA, e 7ì 7j øõ(, ªJ²Çø_œÀíƒ : cqø_ê = a T Ì ª(ininitely ierentiable) í ƒb () í ower series in ( a) Ñ () = c 0 c 1 ( a) c ( a) c 3 ( a) 3, converges or I, (n) () = n(n 1) 1 c n (n1)n 3 c n1 ( a) (n)(n1) 4 3 c n ( a) = c n (n1)! c 1! n1 ( a) 1 (n)! c n ( a) (n! c n ( a), = (n) (a) = c n, ¹ c n = (n) (a) µó () í ower series in ( a) ZªZŸÑ () = (a) (a) 1! ( a) 1 (a) ( a) (n) (a) ( a) n ø_ ower series ÉÊ }Ygí8 -n<,.bbu?v í close orm ø_ % close orm íƒb (), AÐíì D à Bb;bø ÊBó8 -, n?ê () í ower series 5È å,u, ÿ.âz ower series o () í Yg ( D ) v V #ì ower series 5Yg(øì Ý, 1/ ) <cqùå j ø : Êcq ower series "úyg í æ-, â bög (à ratio test ) )ƒ j ù : òqv ower series í close orm, * ícq Ë)ƒ s b ðy¹íiäõ, ÄÑ µsõÿu bö¹ inconclusive í8 FJLSø_ ower series in ( a) Ê} } CJ ( a) n ( íy¹íqõyh, ¹ raius o convergence ó FJBbªJÊø í } } J ( a) n (, y ðy¹íiäõ¹ª Ou, s_. ower series øìbê byg>õý vnªjªws65èí«, syg>õ¹ñhíyg s_ ower series óî, ª JÅÎ (Long Division) O 5,?ª J j²: I (c 0 c 1 c ) = a 0 a 1 a, b 0 b 1 b ¹ (c 0 c 1 c )(b 0 b 1 b ) = (a 0 a 1 a ), Ç(, Usi Ÿáí[b.ìó, 7j) c 0 c 1 c Coyright c 004 by Dr. Mengnien Wu : mwu@mail.tu.eu.tw
Emhases o Calculus Ininite Sequences an Series Page 4 cí Maclaurin b: ln(1 ) arctan = 1 1 3 3 4 4 * = 1 1 3 3 5 5 7 7 * sin = 3 5 7 3! 5! = cos = 1 4 6 4! 6! * 1 C 7V 1 C 7V (1) 7! sin 7V () e = 1 3 4 1! 3! = sinh = 3 5 7 3! 5! 7! * sinh e (3) 4! = e e 7V = e e 7V = cosh = 1 4 4! 6 6! * cosh e â (1)()(3) õ J T í Euler s ormula: e i = cos i sin, i e = 1 BbJ Rû ln(1 ), arctan í erivatives, ]<.Ÿ ln(1 ) = 0 = a (a) = 0 í8 -na) Wà,,5 cos, Bb6ªâ cos = ( ) sin C 7)5: cos = 4 6 8 I =0 C, =.ì)ƒ C = 1. 3!4 5!6 7!8 t VzCì, ÿu"5b"7 ()! 1t à cos = a sin t t VŸíu, a "ú.u 0, 7@vu π 0 t, C arctan = 1t (t) t = () (a) í Ï (UÉÊ t íjbi N.bJÑ a,u 0 à, () ªJ/qËJ ower series [, () = () C í ower series: () à ower series [ý(, }, 7(H/ = a 7)b C J () Ñ n 1 Ÿª, () = n ( (a) ( a) n1 ()( a) n1 á R n() ú a ǃ nÿ n ( (t) ì g(t) e = () ( t) ( t)n1 R n (), g(t) Ê a 5È /ª, / ( a) n1 ú t ǃ nÿ g() = 0 = g(a). ;W Rolle s Theorem (c,ç ), æê ξ Ê a 5È, U) g (ξ) = 0, i.e. [ ] [ n ] 0 = t g(t) (1) (t) = ( t) ( (t) ( t) 1 (n 1)( t)n R n () ( a) n1 [ = ] (n1) (t) ( t) n ( t)n (n 1)R n (). ( a) n1 FJ á, 6ÿuF íïïá R n () = (n1) (ξ) (n 1)! ( a)n1. ( <v, ÛÊÉÌá, ³BóY. Y¹í) à Ê/ õ Ê a í ower series, ¹ () ower series Y¹u ø9. ùáç (Binomial Eansion) ír n ( (a) ( a), ¹ lim R n () = 0, I N {0}. For Z, ( ) :=! =!(! ( 1) ( ) ( 1) ( 1) 1 = ( 1) ( )( 1) ; Coyright c 004 by Dr. Mengnien Wu : mwu@mail.tu.eu.tw
Emhases o Calculus Ininite Sequences an Series Page 5 OuúÝ cbí 7, ( ) :=! ³åí! (! u³<í FJBbJ(ø.yà (!! ( 1) ( r)( r1), 7ZJ Vì ( (!, R: ( ) := ( 1) ( )( 1). I () := (1 ), (0) = 1, = () = (1 ) 1, (0) =, = () = ( 1)(1 ), (0) = ( 1), = ( = ( 1) ( 1)(1 ), ( (0) = ( 1) ( 1)., () í Maclaurin Ç Ñ () = (0) (0) 1 (0) ( (0) 1! = 1 1! 1 ( 1) 1 ( 1) ( 1) i.e. (1 ) = ( ( 0) ( 1) ) ( = (a ) = a (1 a ) = a [( ( 0) ) 1 ( ) ( ) a ( a ) ( ) a ) ] = ( ) 0 a ( ) 1 a 1 ( ) a ( a,..v- N {0} í8 ÄÑOÎ ( íì, ( = 0 or > N {0} FJ ( ué 1 á, 5bJ øíùáçøš I () uø_ n ŸÖá(olynomial o egree n) í Maclaurin Ç ¹uw ; 1/, wú L< a 0 5Ç(Taylor series in a), ªâã Î (Synthetic Division) /q A 'ÀUË, Ä Ñ ( () 0 or > n,.u Maclaurin C Taylor Ç, ÝÖ n 1 á : () := c 0 c 1 1 c c n n = (0) (0) 1 (0) (n) (0) n 1! (à \½ƒ ( (0)Öý, v.}öíú () û Ÿß?) = (a) (a) ( a) 1 (a) ( a) (n) (a) ( a) n 1! Coyright c 004 by Dr. Mengnien Wu : mwu@mail.tu.eu.tw