Числовые и функциональные ряды (Лекции для ИУ), страница 3

tAKOM OBRAZOM D = {x : −1 6 x < 0}.ÔÍ-12ÔÍ-124oPREDELENIE 2.gOWORQT, ^TO FUNKCIONALXNYJ RQD {ak (x)k>1 SHODITSQ RAWNOMERNO NA MNOVESTWEM ⊂ D ⊂ Ω, ESLI (∀ ε > 0) (∃ N (ε) > 1) : |S(x) − Sn (x)| < ε, (∀ n > N (ε)) ∧ (∀ x ∈ M ).ÌÃÒÓÔÍ-127ÌÃÒÓÔÍ-12ÌÃÒÓÌÃÒÓÌÃÒÓÔÍ-12 yÔÍ-12ÔÍ-12ÔÍ-12S(x)Sn (x)S(x) − εx0pRIMERpUSTX Ω = {x ∈ RrIS.30A41x; k=1.xn − xn−1 ; k > 1 0 ;x=0nx ;0<x<11 ;x = 1ÔÍ-12ÔÍ-12ÌÃÒÓ: 0 6 x 6 1}; ak (x) =232nn−1nSn (x) = x + (x − x) + (x − x ) + . .

. + (x − x ) = x ≡0 ; 06x<1I∃lim Sn (x)=S(x)≡.nOTOGDA1 ; x=1n→∞|x|n ; 0 < x < 1|S(x) − Sn (x)| =< ε ⇒ (|x|n < ε) ⇒ n > ln ε/ ln |x| ≡ N (ε, x) ,0; x=1Ix=0T.E ISHODNYJ RQD SHODITSQ, NO NE RAWNOMERNO.ÌÃÒÓ2.ÌÃÒÓS(x) + εpRIZNAK wEJER[TRASSA.ÌÃÒÓÌÃÒÓeSLI NA MNOVESTWE M ⊂ D DLQ FUNKCIONALXNOGO RQDA {ak (x)}k>1SU]ESTWUET MAVORANTA {bk }k>1 , T.E. |ak (x)| 6 bk , ∀ k > 1, ∀ x ∈ M I ZNAKOPOLOVITELXNYJ ^ISLOWOJRQD {bk }k>1 SHODITSQ, TO RQD {ak (x)}k>1 SHODITSQ NA MNOVESTWE M ⊂ D ABSOL@TNO I RAWNOMERNO.dOKAZATELXSTWO.pUSTXWYPOLNENYUSLOWIQTEOREMY.tOGDA|ak (x)| < bk , (∀ k > 1) ∧ (∀ x ∈ M ⊂ D). a TAK KAK RQD {bk }k>1 SHODITSQ, TO PO PRIZNAKUSRAWNENIQ W KAVDOJ TO^KE x ∈ M ISHODNYJ FUNKCIONALXNYJ RQD SHODITSQ ABSOL@TNO.4pUSTX Rn (x) = an+1 (x) + .

. . + an+m (x) + . . . - n-YJ OSTATOK ISHODNOGO FUNKCIONALXNOGO RQDA I4ÔÍ-12ÔÍ-12σn+m (x) = an+1 (x) + . . . + an+m (x) - m-AQ ^ASTNAQ SUMMA \TOGO n-GO OSTATKA.tAK KAK RQD {ak (x)}k>1 SHODITSQ NA M ⊂ D ABSOL@TNO, TO ON SHODITSQ NA M ⊂ D I σnm (x) → Rn (x)PRI m → ∞ ∀x ∈ M ⊂ D. a TAK KAK ZNAKOPOLOVITELXNYJ ^ISLOWOJ RQD {bk }k>1 SHODITSQ, TO∞X(∀ ε > 0)(∃ N (ε) > 1) : bn+1 + . . . + bn+m 6bn+k < ε, ∀ n > N (ε) ∧ ∀ m > 1. tAKIM OBRAZOM,k=1(∀n > N (ε))∧(∀x ∈ M )∧(∀ m > 1) |σnm (x)| 6 |an+1 (x)|+. . .+|an+m (x)| 6 bn+1 +. . .+bn+m < ε, ∀m > 1,T.E.

|Rn (x)| = lim σnm (x)| < ε, (∀ n > N (ε)) ∧ (∀ x ∈ M ), ^TO I TREBOWALOSX DOKAZATX.m→∞pRIMER 3.4sWOJSTWA RAWNOMERNO SHODQ]IHSQ RQDOWÌÃÒÓÌÃÒÓrQD S OB]IM ^LENOM an (x) = xn /n2 PRI |x| 6 1 SHODITSQ RAWNOMERNO, TAK KAK WRASSMATRIWAEMOM SLU^AE |ak (x)| 6 1/k 2 , ∀ k > 1 I ZNAKOPOLOVITELXNYJ ^ISLOWOJ RQD {1/k 2 }k>1SHODITSQ.ÔÍ-12ÔÍ-121.

eSLI ∀ k > 1 FUNKCIQ ak (x) NEPRERYWNA NA M ⊂ D I NA M RAWNOMERNO SHODITSQ FUNKCIONALXNYJ RQD {ak (x)}k>1 , TO EGO SUMMA S(x) NEPRERYWNA NA M (SM. PRIMER 2).2. sUMMU S(x) RAWNOMERNO SHODQ]EGOSQ NA M ⊂ D FUNKCIONALXNOGO RQDA {ak (x)}k>1 NEPRERYWNYH (DIFFERENCIRUEMYH) NA M ⊂ D FUNKCIJ ak (x), k > 1, MOVNO PO^LENNO INTEGRIROWATX (DIFFERENCIROWATX) NA M .oPREDELENIE 3. eSLI a (x) = α· (x − x0 )k ; ∀ k > 0, GDE αk – WE]ESTWENNOE ILI KOMPLEKSNOE ^ISLO,TO FUNKCIONALXNYJ RQD {αk · (x − x0 )k }k>0 NAZYWA@T STEPENNYM RQDOM.kzAME^ANIE 1.

zAMENOJ t = x − x40STEPENNOJ RQD {αk · (x − x0 )k }k>0 WSEGDA MOVET BYTX PRIWEDENK STANDARTNOMU WIDU {αk tk }k>0 .ÔÍ-128ÌÃÒÓkÌÃÒÓÔÍ-12ÌÃÒÓÌÃÒÓÔÍ-12ÌÃÒÓÌÃÒÓÔÍ-12oPREDELENIE 4. iNTERWALOM SHODIMOSTI WE]ESTWENNOGO STEPENNOGO RQDA {αÔÍ-12ÔÍ-12k x }k>0 NAZYWA@T MNOVESTWO (−R; +R) ⊂ R1 , W KAVDOJ TO^KE KOTOROGO ISHODNYJ RQD SHODITSQ ABSOL@TNO I RASHODITSQPRI |x| > R, GDE R NAZYWA@T RADIUSOM SHODIMOSTI.kzAME^ANIE 2.sU]ESTWOWANIE INTERWALA SHODIMOSTI SLEDUET IZ TEOREMY aBELQ, KOTORU@ MYSFORMULIRUEM I DOKAVEM DLQ KOMPLEKSNYH STEPENNYH RQDOW.

nA DANNOM \TAPE OGRANI^IMSQ SLEDU@]IMI FORMALXNYMI RASSUVDENIQMI.pO d’aLAMBERA, FORMALXNO IMEEM: PRIZNAKUÌÃÒÓÌÃÒÓ ak+1 (x) = |x| lim |αk+1 | < 1, T.E. RQD {αk · xk }k>0 SHODITSQ ABSOL@TNO PRI |x| < R = lim αk lim k→∞k→∞ |αk |k→∞ αk+1 ak (x)I RASHODITSQpPRI |x| > R – SM. TAK VE PRIMER 1. eSLI ISPOLXZOWATX PRIZNAK kO[I S RADIKALOM, TOR = 1/ lim k |αk |.k→∞ k xkk+2x4. eSLI αk (x) =, TO R = lim= 1.

tAKIM OBRAZOM RQDSHODITSQk→∞ k + 1k+1k + 1 k>0(−1)k1I ak (+1) =, TO ESTX PRIABSOL@TNO PRI |x| < 1 I RASHODITSQ PRI |x| > 1. ak (−1) =k+1k+1x = −1 RASSMATRIWAEMYJ RQD SHODITSQ USLOWNO, A PRI x = +1 ON RASHODITSQ.(k + 1)!5. eSLI ak (x) = xk /k!, TO αk = 1/k! I R = lim= ∞ I FUNKCIONALXNYJ RQDk→∞k!{xk /k!}k>0 SHODITSQ ABSOL@TNO PRI |x| < R = ∞.pRIMERoSNOWNYE UTWERVDENIQ O STEPENNYH RQDAHÔÍ-12ÔÍ-12pRIMERÔÍ-12ÔÍ-12ÌÃÒÓÌÃÒÓI.

eSLI R > 0 – RADIUS SHODIMOSTI STEPENNOGO RQDA {αk xk }k>0 , TO \TOT RQD BUDET SHODITXSQRAWNOMERNO NA L@BOM OTREZKE [−r; +r] ⊂ (−R; +R).dOKAZATELXSTWO. eSLI 0 < r < R, TO PRI |x| = r ISHODNYJ RQD SHODITSQ ABSOL@TNO, T.E.SHODITSQ ZNAKOPOLOVITELXNYJ ^ISLOWOJ RQD {|αk | r k }. nO TOGDA PRI L@BOM x ∈ [−r; r] IMEEM |x| 6 r,T.E. |αk xk | = |αk | |x|k 6 |αk | r k I OSTALOSX WOSPOLXZOWATXSQ T. wEJER[TRASSA.II. sUMMU S(x) cTEPENNOGO RQDA {αk xk }k>0 S RADIUSOM SHODIMOSTI R > 0 MOVNO PO^LENNO DIFFERENCIROWATX W INTERWALE SHODIMOSTI PROIZWOLXNOE ^ISLO RAZ. pRI \TOM WSE PROIZWODNYE STEPENNYERQDY IME@T TOT VE RADIUS SHODIMOSTI R.dOKAZATELXSTWO.sOGLASNO UTWERVDENI@ I ISHODNYJ RQD SHODITSQ RAWNOMERNO NA L@BOM OTREZKE [−r; r] ⊂ (−R; R) I EGO SUMMU MOVNO PO^LENNO DIFFERENCIROWATX.pUSTXk{βk x }k>0 – PROIZWODNYJ STEPENNOJ RQD, T.E. βk = (k + 1) αk+1 , ∀ k > 0.

tAKIM OBRAZOM|αk |k |αk ||βk−1 |≡ lim= lim, ^TO I TREBOWALOSX DOKAZATX.R = limk→∞ |αk+1 |k→∞ (k + 1)|αk+1 |k→∞ |βk |!00∞∞XX111.xk =PRI |x| < 1. tOGDA=xk , TO ESTX PRI |x| < 1(1 − x)1−xk=0k=0∞∞XX11=k xk−1 . pRI |x| < 1 ⇐⇒k xk−1 =.2(1 − x)(1 − x)2k=0k=1pOLAGAQ n = k − 1, T.E k = n + 1, POLU^AEM:!00∞∞∞XXX11nn=n(n + 1) xn−1 .=(n + 1)x PRI |x| < 1.

dALEE IMEEM:=(n + 1)x2(1 − x)2(1−x)n=0n=0n=0∞X2tAKIM OBRAZOMn(n + 1) xn−1 =, PRI |x| < 1. pOLAGAEM m = n − 1, T.E. n = m + 1. tOGDA(1 − x)3n=1∞X2PRI |x| < 1 I T.D.(m + 1)(m + 2)xm =(1 − x)3m=0ÔÍ-12ÔÍ-12ÌÃÒÓÌÃÒÓpRIMERÌÃÒÓÔÍ-129ÌÃÒÓÔÍ-12ÌÃÒÓÌÃÒÓÌÃÒÓÔÍ-12ÔÍ-12ÌÃÒÓIII. sUMMU STEPENNOGO RQDA {αk x }k>0 S RADIUSOM SHODIMOSTI R > 0 MOVNO PO^LENNO INTEGRIROWATX W INTERWALE SHODIMOSTI PROIZWOLXNOE ^ISLO RAZ. pRI \TOM POLU^ENNYJ STEPENNOJ RQD IMEETTOT VE RADIUS SHODIMOSTI R.dOKAZATELXSTWO. sOGLASNO I ISHODNYJ STEPENNOJ RQD SHODITSQ RAWNOMERNO NA L@BOM OTREZKE[−r; r] ⊂ (−R; R) I DOPUSTIMO PO^LENNOE INTEGRIROWANIE EGO SUMMY. pUSTX {βn xn }n>0 – NOWYJSTEPENNOJ RQD, POLU^ENNYJ PUTEM INTEGRIROWANIQ ISHODNOGO RQDA PRI |x| < R:αk xk dxT.E.αk−1k=Iβk+1=αk.k+1nOTOGDAÌÃÒÓβkÌÃÒÓ0αk xk+1,k+1=ÔÍ-12ZxÔÍ-12k|αk |(k + 1) |βk+1 ||βk+1 |= lim= lim, ^TO I TREBOWALOSX DOKAZATX.k→∞ |αk+1 |k→∞ (k + 2) |βk+2 |k→∞ |βk+2 |ZxZx X∞∞Xdx1k2.PRI |x| < 1.

tOGDA=xk dx; |x| < 1, TO ESTXx =1−x1−xk=0k=0R = limpRIMER0xÔÍ-12kx dx =k=0 0PRI |x| < 1.∞Xk=00∞X1 n1k+1x , |x| < 1. eSLI n = k + 1, TOx = − ln(1 − x)k+1nn=1ÔÍ-12− ln(1 − x) =∞ ZXrQD tEJLORAÌÃÒÓn→∞RQD {[f (k) (x0 )/k!] (x − x0 )k }k>0 QWLQETSQ NA (a; b) SHODQ]IMSQ, A f (x) QWLQETSQ EGO SUMMOJ:k=1k!(x − x0 )k ; x, x0 ∈ (a; b).ÔÍ-12ÔÍ-12f (x) = f (x0 ) +∞Xf (k) (x0 )ÌÃÒÓI. pUSTX FUNKCIQ y = f (x) OPREDELENA I BESKONE^NO DIFFERENCIRUEMA W INTERWALE (a; b) ⊂ R1 .eSLI x0 ∈ (a; b), TO PO FORMULE tEJLORA IMEEM:nXf (k) (x0 )f (x) = f (x0 ) +(x − x0 )k + Rn (x; x0 ); ∀ x ∈ (a; b)k!k=1pRI \TOM, ESLI DLQ L@BOGO FIKSIROWANNOGO x ∈ (a; b) SU]ESTWUET lim Rn (x, x0 ) = 0, TO STEPENNOJf (k) (x0 )k(x − x0 )NAZYII. dLQ BESKONE^NO DIFFERENCIRUEMOJ FUNKCII f (x) STEPENNOJ RQDk!k>0WA@T EE RQDOM tEJLORA WNE ZAWISIMOSTI OT HARAKTERA SHODIMOSTI \TOGO RQDA.

eSLI x0 = 0, TO RQDtEJLORA NAZYWA@T RQDOM mAKLORENA.III. dLQ ANALIZA SHODIMOSTI RQDA tEJLORA MOVNO ISPOLXZOWATX STANDARTNYE PRIEMY TEORIISTEPENNYH RQDOW, NO MOVNO ISPOLXZOWATX I SLEDU@]U@ TEOREMU.ÌÃÒÓ1PRI NEKOTORYH B > 0 I C > 0 WYPOLNQETSQ NERAWENSTWOÌÃÒÓtEOREMA.

eSLI W INTERWALE (a; b) ⊂ RÔÍ-12|x − x0 |n+1(n + 1)!sup |f (n+1) (x∗ )| 6a<x∗ <b|x − x0 |n+1· C · B n+1 · (n + 1)! = C · {B|x − x0 |}n+1 → 0n→∞(n + 1)!TOGDA I TOLXKO TOGDA, KOGDA |x − x0 | < 1/B.ÔÍ-1210ÌÃÒÓ|Rn (x, x0 )| 6ÔÍ-121sup |f (k) (x)| 6 C; ∀ k > 1, TO RQD tEJLORA, BESKONE^NO DIFFERENCIRUEMOJ NA (a; b) FUNKCII· k! a<x<bf (x), SHODITSQ WO WSEH TO^KAH x ∈ (a; b), DLQ KOTORYH |x − x0 | < 1/B.dOKAZATELXSTWO. pUSTX WYPOLNENY USLOWIQ TEOREMY, TOGDA(x − x0 )n+1 (n+1) ∗Rn (x, x0 ) =f(x ); x < x∗ < x0 ILI x0 < x∗ < x.(n + 1)!tAKIM OBRAZOM,BkÌÃÒÓÔÍ-12ÌÃÒÓÔÍ-12ÌÃÒÓsTEPENNYE RQDY W CtEOREMA aBELQ. eSLI STEPENNOJ RQD {aÔÍ-12ÌÃÒÓÔÍ-12ÔÍ-12ÌÃÒÓz k }k>0 S KOMPLEKSNYMI ^LENAMI SHODITSQ PRI NEKOTOROMz = z1 6= θ, TO ON ABSOL@TNO SHODITSQ ∀z : |z| < |z1 |.