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7/23/2019 Comment montrer qu'une suite est convergente ?
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I c b d g R u c t g s „
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m n g s g t J k k j a g s „
d t t p < / / g x m s ? e j t d
. i r g g . i r
/
Emktrgr qu–ukg suctg rãgaag gst bmkvgrfgktg.
_jppga ng bmurs
Nãkctcmk
Wkg suctg (uk)k∄K gst nctg bmkvgrfgktg vgrs uk rãga ℒ, sc tmut cktgrvjaag muvgrt bmktgkjkt ℒ bmktcgkt tmus ags tgregs
ng aj suctg è pjrtcr n–uk bgrtjck rjkf k8. Wkg suctg quc kg bmkvgrfg pjs gst nctg ncvgrfgktg.
Xrmprcãtãs
• Rc ukg suctg bmkvgrfg, jamrs sj acectg gst ukcqug < ace uk 5 ℒ.• Rc (uk) bmkvgrfg vgrs ℒ, sc (vk) bmkvgrfg vgrs ℒ ‿, gt sc, è pjrtcr n–uk bgrtjck rjkf < uk ≮ vk, jamrs ℒ ≮ ℒ‿.• Rc (uk) tgkn vgrs +∛, gt sc, è pjrtcr n–uk bgrtjck rjkf < uk ≮ vk, jamrs (vk) tgkn vgrs +∛.• Rc, è pjrtcr n–uk bgrtjck rjkf < |uk ∟ ℒ| ≮ vk, gt sc aj suctg (vk) bmkvgrfg vgrs ℒ‿, jamrs (uk) bmkvgrfg vgrs (ℒ + ℒ‿).• Rmct (uk)k≧k8 ukg suctg nãkcg pmur tmut k ≧ k8 pjr uk 5 i (k). Rc ace
x↟+∛i (x) 5 ℒ, jamrs ace
k↟+∛uk 5 ℒ.
^dãmräegs
• Rc ngux suctgs (uk) gt (vk) bmkvgrfgkt vgrs uk rãga ℒ, gt sc, è pjrtcr n–uk bgrtjck rjkf < uk ≮ wk ≮ vk, jamrs (wk)bmkvgrfg jussc vgrs ℒ (^dãmräeg ngs fgknjregs).• Wkg suctg brmcssjktg gt ejlmrãg gst bmkvgrfgktg (^dãmräeg ng aj bmkvgrfgkbg emkmtmkg).• Wkg suctg nãbrmcssjktg gt eckmrãg gst bmkvgrfgktg (^dãmräeg ng aj bmkvgrfgkbg emkmtmkg).• Rmct (uk)k≧k8
ukg suctg nãkcg pjr rãburrgkbg pmur tmut k ≧ k8 pjr uk+: 5 i (uk). Rc bgttg suctg gst bmkvgrfgktggt sc aj imkbtcmk i gst bmktckug gk ℒ, jamrs sj acectg ℒ gst ukg smautcmk ng a–ãqujtcmk ℒ 5 i (ℒ).
Eãtdmng
Xmur sjvmcr sc ukg suctg bmkvgrfg, vmus a–jurgz bmeprcs, ca ijunrj gssjygr « n–gkbjnrgr ° bgttg suctg.Mk pmurrj nmkb ãtuncgr sgs omrkgs pucs s–jcngr ngs prmprcãtãs gt ngs tdãmräegs rjppgaãs bc-ngssus pmur bmkbaurg.
Gxgrbcbg : (Xmaykãscg ?8:2)
Mk bmkscnärg aj suctg (uk)k∄K nãkcg pjr u8 5 :
?
gt tgaag qug pmur tmut gktcgr kjturga k, uk+: 5 2uk
: + ?uk
.
Mk jnegt qug pmur tmut gktcgr kjturga k, 8 1 uk 1 :.Nãemktrgr qug aj suctg (uk) gst brmcssjktg. Gk nãnucrg qug aj suctg (uk) bmkvgrfg.
Gxgrbcbg ? (Xmkncbdãry ?8:2)
Rmct ags suctgs (C k) gt (L k) nãkcgs pmur tmut k ∄ K pjr < C k 5
:8
g∟kx
: + x nx gt L k 5
:8
g∟kx
(: + x)? nx.
:. Emktrgr qug pmur tmut gktcgr k ≧ 8 gt pmur tmut rãga x ∄ P80 :U < 8 ≮ g∟kx
(: + x)? ≮
g∟kx
: + x ≮ g∟kx.
?. Emktrgr qug ags suctgs (C k) gt (L k) smkt bmkvgrfgktgs gt nãtgreckgr agur acectg.
Gxgrbcbg :
:. Bmeeg pmur tmut k ∄ K, uk ; 8, pmur ãtuncgr agsvjrcjtcmks ng aj suctg, mk pgut bmepjrgr uk+:
uk
è : <
uk+:
uk
52uk
:+?uk
uk
5 2
: + ?uk
ejcs pmur tmut k ∄ K < uk 1 :
⇞ ?uk 1 ? ⇞ : + ?uk 1 2 ⇞ : 1 2:+?uk
bjr : + ?uk ; 8.
Ickjagegkt aj suctg (uk) gst brmcssjktg.
?. Aj suctg (uk) gst brmcssjktg gt ejlmrãg pjr : 0 gaagbmkvgrfg nmkb vgrs ℒ ≮ : n–jpräs ag tdãmräeg ng aj bmkvgr-fgkbg emkmtmkg.Jttgktcmk < kg pjs bmkbaurg qug aj acectg vjut : ! Gkggt, mk j jussc uk 1 :8 ejcs mk kg pgut pjs bmkbaurgkmk paus qug sj acectg vjut :8. Xmur bjabuagr aj acectg caijut nmkb ecgux gkbjnrgr uk mu motgkcr smk gxprgsscmk.
Gxgrbcbg ?
:. x ≧ 8 ⇟ : + x ≧ : ⇟ ::+x
≮ ::
mr g∟kx ; 8 nmkb pmur
tmut k ∄ K < g∟kx
:+x≮ g∟kx.
Ng paus : + x ≧ : ⇟ (: + x)? ≧ :(: + x) ⇟ :(:+x)? ≮ :
:+x
nmkb pmur tmut k ∄ K < g∟kx
(:+x)? ≮ g
∟kx
:+x.
Gkk ca gst ãvcngkt qug pmur tmut k ∄ K < g∟kx
(:+x)? ≧ 8
nmkb mk motcgkt ocgk < 8 ≮ g∟kx
(:+x)? ≮ g
∟kx
:+x≮ g∟kx.
?. Xjr cktãfrjtcmk sur a–cktgrvjaag P80 :U ngs ckãfjactãs prã-bãngktgs, mk motcgkt pjr brmcssjkbg ng a–cktãfrjag, aj rgaj-
tcmk pmur tmut k ∄ K < 8 ≮ L k ≮ C k ≮
∟ :k
g∟kx:8
.
Rmct < 8 ≮ L k ≮ C k ≮ ∟ :k
g∟k ∟ :
. Mr ace
k↟+∛g∟k 5 8
nmkb pjr prmnuct ng acectgs acek↟+∛ ∟ :
k
g∟k ∟ :
5 8.
N–jpräs ag tdãmräeg ngs fgknjregs, (C k) gt (L k)bmkvgrfgkt vgrs 8.
