1 / 13

TOAÙN 4 CHUOÃI VAØ PHÖÔNG TRÌNH VI PHAÂN BAØI 1: CHUOÃI SOÁ (PHAÀN 2)

BOÄ MOÂN TOAÙN ÖÙNG DUÏNG - ÑHBK -------------------------------------------------------------------------------------. TOAÙN 4 CHUOÃI VAØ PHÖÔNG TRÌNH VI PHAÂN BAØI 1: CHUOÃI SOÁ (PHAÀN 2) TS. NGUYEÃN QUOÁC LAÂN (3/2006).

nicola
Download Presentation

TOAÙN 4 CHUOÃI VAØ PHÖÔNG TRÌNH VI PHAÂN BAØI 1: CHUOÃI SOÁ (PHAÀN 2)

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. BOÄ MOÂN TOAÙN ÖÙNG DUÏNG - ÑHBK------------------------------------------------------------------------------------- TOAÙN 4 CHUOÃI VAØ PHÖÔNG TRÌNH VI PHAÂN • BAØI 1: CHUOÃI SOÁ (PHAÀN 2) • TS. NGUYEÃN QUOÁC LAÂN (3/2006)

  2. NOÄI DUNG--------------------------------------------------------------------------------------------------------------------------------- 5- TIEÂU CHUAÅN D’ALAMBERT (TYÛ SOÁ), COÂSI 6- CHUOÃI DAÁU BAÁT KYØ. T/CHUAÅN HOÄI TUÏ TUYEÄT ÑOÁI 7- CHUOÃI ÑAN DAÁU. TIEÂU CHUAÅN LEBNITZ

  3.  Daõy toång rieâng {Sn}: Chuoãi döông un, un > 0  n  N0  döông hoäi tuï khi vaø chæ khi bò chaën: Daáu hieäu so saùnh 1: un, vn vôùi 0 < un  vn,  n  N0 vn (chuoãi lôùn) htuï  un (nhoû) htuï: un (nhoû) ph.kyø  vn (lôùn) ph.kyø: VD: Khaûo saùt söï hoäi tuï cuûa caùc chuoãi CHUOÃI DÖÔNG--------------------------------------------------------------------------------------------------------------------------

  4. “Ñoaùn” tính hoäi tuï cuûa chuoãi: Chuoãi ñieàu hoaø (Rieman) Chuoãi Rieman hoäi tuï   > 1 So saùnh vôùi chuoãi Rieman CHUOÃI ÑIEÀU HOAØ (CHUOÃI RIEMAN)--------------------------------------------------------------------------------------------------------------------------------------- Tính toång rieâng. Laäp baûng giaù trò {n Sn}  Tính chaát hoäi tuï:

  5. Chuoãi döông un, vn (töø chæ soá N0). Neáu toàn taïi giôùi haïn :2 chuoãi cuøng baûn chaát hoäi tuï k=0  un < vn n  N1 & k= un > vn: Aùp duïng so saùnh 1 VD: Khaûo saùt söï hoäi tuï cuûa caùc chuoãi: Nguyeân taéc: Duøng töông ñöông, so saùnh un vôùi chuoãi 1/n (töông töï tích phaân suy roäng!). Moät soá tröôøng hôïp coù theå aùp duïng khai trieån Mac – Laurint theo x = 1/n vôùi un DAÁU HIEÄU SO SAÙNH 2---------------------------------------------------------------------------------------------------------------------------------------

  6. Chuoãi döông un coù giôùi haïn tyû soá: a/ d < 1: Hoäi tuï b/ d > 1: Phaân kyø c/ d = 1: Chöa keát luaän! Daáu hieäu D’Alambert duøng cho nhöõng chuoãi coù tyû soá un+1/un “ñôn giaûn”: chuoãi chöùa giai thöøa hoaëc muõ VD: Khaûo saùt un: • d = 1 hoaëc Khoâng  lim un+1/un: chuoãi coù theå hoäi tuï laãn phaân kyø. Ví duï: TIEÂU CHUAÅN D’ALAMBERT (TYÛ SOÁ) --------------------------------------------------------------------------------------------------------------------------------------- • d = 1 & un+1/un  1  n  N0: chuoãi un phaân kyø (ñkieän caàn!)

  7. Chuoãi döông un vaø  giôùi haïn caên: q < 1: Hoäi tuï q > 1: Phaân kyø q = 1: chöa keát luaän TC Coâsi: Chuoãi chöùahaøm muõ (hoaëc luyõ thöøa baäc n) VD: q = 1 hoaëc Khoâng  lim (un)1/n : khoâng theå keát luaän q = 1 vaø (un)1/n  1: Chuoãi phaân kyø (ñieàu kieän caàn)! TIEÂU CHUAÅN COÂSI (TIEÂU CHUAÅN CAÊN)---------------------------------------------------------------------------------------------------------------------------

  8. Chuoãi soá un, un – daáu baát kyø  Khoâng ñöôïc pheùp aùp duïng tieâu chuaån so saùnh 1 – 2, D’Alambert laãn Coâsi hay bò chaën! Chuoãi |un| hoäi tuï  Chuoãi un hoäi tuï & goïi laø hoäi tuï tuyeät ñoái VD: (): Sai.  |un| phaân kyø nhöng un vaãn hoäi tuï: Baùn hoäi tuï •  |un| hoäi tuï  un hoäi tuï: hoäi tuï tuyeät ñoái Keát luaän: •  |un| phaân kyø, un hoäi tuï: baùn hoäi tuï • un phaân kyø   |un| phaân kyø CHUOÃI DAÁU BAÁT KYØ --------------------------------------------------------------------------------------------------------------------------------

  9.  |un| phaân kyø (D’Alambert)  un phaân kyø (ñkieän caàn!)  |un| phaân kyø (vôùi TC Coâsi)  un phaân kyø (ñkieän caàn!) Khaûo saùt  |un| , neáu aùp duïng tieâu chuaån D’Alambert (Coâsi) •  |un| hoäi tuï   unhoäi tuï tuyeät ñoái • |un| phaân kyø  un phaân kyø (ñieàu kieän caàn) TIEÂU CHUAÅN D’ALAMBERT (COSI) VÔÙI CHUOÃI DAÁU ------------------------------------------------------------------------------------------------------------------------------

  10.  (–1)n-1bn = b1 – b2 + b3 – … (bn > 0): chuoãi ñan daáu Tchuaån Lebnitz: Neáu daõy {bn} giaûm: b1 > b2 > … > bn > … vaø tieán veà 0: limbn = 0  chuoãi ñan daáu (–1)n-1bn hoäi tuï. Kyõ thuaät haøm soá chöùng minh daõy giaûm. VD: CHUOÃI ÑAN DAÁU-----------------------------------------------------------------------------------------------------------------------------

  11. b1 -b2 +b3 -b4 +b5 -b6 s3 s1 s2 s5 s4 s6 s 0 MINH HOAÏ HOÄI TUÏ LEBNITZ-------------------------------------------------------------------------------------------------------------------------------

  12. Daõy {bn}: b1 > b2 > … > …  Trò tuyeät toång rieâng Sn  | b1 |: | b1 – b2 + b3 – … + bn |  | b1 | & Sn cuøng daáu b1. Töông töï, öôùc löôïng phaàn dö: |Rn| = |bn+1–bn+2 + …| |bn+1 | VD: Chöùng minh chuoãi sau hoäi tuï veà toång S. Tính gaàn ñuùng S vôùi sai soá 10-1 • Chuoãi ñan daáu. Xaùc ñònh coâng thöùc bn vaø kieåm tra daõy  • Thieát laäp lim bn = 0  Hoäi tuï theo tieâu chuaån Lebnitz • Xaáp xæ S  Sn, sai soá Rn coù | Rn |  bn+1   = 10-1  n  ? ÖÔÙC LÖÔÏNG TOÅNG CHUOÃI ÑAN DAÁU--------------------------------------------------------------------------------------------------------------------------------

  13. CHUOÃI an: TOÅNG VOÂ HAÏN = limn TOÅNG RIEÂNG Sn   giôùi haïn: HOÄI TUÏ ÑIEÀU KIEÄN PHAÂN KYØ:  Chuoãi an PHAÂN KYØ DÖÔNG: an, an  0 [Hoäi tuï  Bò chaën]  So saùnh 1, 2. D’Alambert, Coâsi DAÁU BAÁT KYØ: Hoäi tuï tuyeät ñoái: |an| hoäi tuï  an hoäi tuï ÑAN DAÁU:(1)nbn, bn > 0 {bn} giaûm, lim bn = 0  Hoäi tuï OÂN TAÄP CHUOÃI SOÁ----------------------------------------------------------------------------------------------------------------------------------

More Related