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Publié par | MATHS-LYCEE.FR |
Publié le | 15 octobre 2015 |
Nombre de lectures | 112 |
Langue | Français |
Extrait
❉✈❡r✐♦✲✶✸✿❙✉✐st❡
✕
▼❆❚❍❙✲▲Y❈❊❊✳❋❘
❚❙r♦✐✈❡❉✲
❊①❡r❝❝✐❡✶
❈❤❛♣❡rt✐✿✶
❉❙ ✶✲✸ ✿ ❙✉✐t❡s t②♣❡ ❇❆❈
❙✉✐t❡s
♦✬▲❜❥t❡❞❡t❡❝①❡❡r❝❝✐❡t❡s❞t✉✬é❞✐r❡❛❧✉s❡t✐(un)✜é❞✉r❡s♥✐N♣❛r
1 7
un+1=un+✳
2un
❖♥❛❞♠❛❡tq❡✉♦♣r✉♦tt✉♥❡✐tr❡t✉♥❛❡r❧rn✱un>0✳
WWW✳▼❆❚❍❙✲▲Y❈❊❊✳❋❘
✳✶✐sé❞♥❖r❛♣❡♥❣f❧♥✐❡tr✉✬❧❡❧✈r❧❛❝t✐♦❛❢♦♥♥✐❡s♥❞é✜+]0 ;∞[♣r❛
WWW✳▼❆❚❍✲❙▲Y❈❊✳❊❋❘
❈❤❛rt✐♣❡✿✶
é❉♠tr♦♥❡rq✉❡❧❛❢♦♥♦✐t❝♥
❊♥❞é❞✉❡r✐❡✉q♣♦r✉♦tt✉❡✐r❡♥t♥❛❡✉tr❧
☛♥t✉♦✐❙❧♦
✿
f♦❞♥é❝r✐t❡✐❞♦❡♥❢s❡ss♦s♠✉♠✈s❛t❜❧❧❛❡r
f✉♠✳❞❛❡♠t✉♥♠✐♥✐♠
√
n✱un>7✳
]0; +∞[
❞♦♥❝ts❡ré❞❜✈❛✐❡❧s✉r]0; +∞[✳
2
1 71x−7
′
f(x1) =−=
2 2
2x2x
2′2
x >0sr✉]0; +∞[♦❞❝♥f(x)st❡s✐❞✉❡❞❣♥❡x−7
√
2
❙r✉]0; +∞[✱x−7 = 0r♣❡✉♦♥♥❧✉s❛✬x= 7
WWW✳▼❆❚❍❙✲▲Y❈❊❊✳❋❘
❖♥❛❞♦♥✭❝s❛s♥❧❡s✐❧♠✮s❡t✐✿
❙✉✐t❡s
P❛❣❡ ✶✴✹
✭❞✉ré❡ ✻✵♠♥✮
u0= 3❡r✉t❧❡✉tr♣♦✉tt♦t✐❡♥♥❛❡r
1 7
f(x) =x+✳
2x
▼❆❚❍✲❙▲Y❈❊❊✳❋❘
✭✽♣♦♥✐ts✮
❡tr♠✐♥❛❧❡❙
n✱
WWW✳▼❆❚❍✲❙▲Y❈❊❊❋✳❘
✕
❉♦✈❡r✐✸✲✶✿❙✐✉t❡s
❉❡♦✈r✐✲✶✸✿❙✉s❡t✐
✕
▼❆❚❍❙✲▲Y❈❊❊✳❋❘
❚❙✲❉r✐♦✈❡
WWW✳▼❆❚❍✲❙▲Y❈❊❊✳❋❘
❈❤❛♣rt❡✐✿✶
✳✷
❈❤❛❡rt✐♣✿✶
√
❞♦❝♥❧❡♠✐♥✐♠✉♠❡❞f❡tsf7✳
√ √
1 7
f+( 7)= 7√
2 7
!
√
√
1 77
= 7+√ √
2 77
√ √
1
= 7+ 7
2
√
= 7
1 7
❖♥ ❛un+1=un+ =f(un)
2un
√
❝❡✈❛f(un)≥7
W√WW✳▼❆❚❍❙✲▲Y❈❊❊✳❋❘
❞♦♥❝un≥7t♥❡✐♥rt❛r✉❧❡♣t❡♦✉rt♦✉n✳
➱t✉❞✐❡r❧❡s✈❛r✐t♦✐❛s♥❡❞❛❧✉st✐❡
(un)✳
☛❙✉t♦❧♥✐♦✿
1 7
un+1−un=un+−un
2un
1 7
=un+−2un
2un
1 7
=−un
2un
2
1 7−u
n
=
2un
❝ts❡❞✉❣✐s❡♥❡❞
❖♥❛❞♠t❡✉q❡un>0❞♦♥un+1−un
√
WWW✳▼❆❚❍❙✲▲Y❈❊❊✳❋❘
2 2
♦run≥7❞♦♥❝u≥7t❡7−u≤0
n n
❞♦♥❝un+1−un≤0
❞♦❝♥❛❧❡✐✉st
❙✉✐t❡s
(un)❡♦❝r❞ésts❡t✳♥❛✐
P❛❣❡ ✷✴✹
❙✉✐t❡s
2
7−u✳
n
▼❆❚❍✲❙▲Y❈❊❊✳❋❘
♠r❡t✐♥❛❧❡❙
WWW✳▼❚❆❍✲❙▲Y❈❊✳❊❋❘
✕
❉✈❡r✐♦✶✲✸✿❙✉✐t❡s
❉❡♦✈r✐✲✶✸✿❙s❡t✐✉
✕
▼❚❆❍❙✲▲Y❈❊❊✳❋❘
❚❙r♦✐❡✈✲❉
✸✳
✳✹
❊①❡r❝✐❝❡ ✷
▲❛s✉t❡✐
☛❙✉t♦❧♥✐♦
▲❛✐❧♠❡t✐
❊♥❞❞é✐✉r❡
(u)t♥❡❣❄❡♦❝❡❧r❡✈♥t✲❡❧❡s
n
✿
❝é♦r✐st♥❡❡♠t♥✐❛
(un)t❞❡s♦ré❡♣❛r
☛❙♦❧♥t✉♦✐
❞♦♥❝
(un)❡r❣❡♥t❡✳❡st❝♦♥✈
ℓ❞❡ ❝❡t❡✉t❡st❡q❧❧s✉❡❡❡✐t
limun✳
n→+∞
✿
❈❤❛♣❡✐rt✶✿
√
7
ℓ=f(ℓ)✳
√ √
❖♥ér♦st✉✉sr[ 7;+∞[❡q✉✐s♣✉un≥7
1 7
f(ℓ) =ℓ⇐⇒ℓ=ℓ+
2ℓ
7
⇐⇒2ℓ=ℓ+
ℓ
⇐W⇒ℓW=W✳▼❆❚❍❙✲▲Y❈❊❊✳❋❘
7
ℓ
2
⇐⇒ℓ= 7
√
⇐⇒ℓ= 7r❛❝ℓ >0
❞♦❝♥
√
limun= 7
n→+∞
❙✐✉t❡s
❖♥♦❝✐s♥❡❞èr❧❡ss✉❡✐st(un)t❡(vn)✉rss❡✐♥✜é❞N♣❛r ✿
un+vnun+1+vn
u0= 3t❡un+1=❡tv0= 4❡tvn+1=✳
2 2
✳✶❈❛