2
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2015
Le téléchargement nécessite un accès à la bibliothèque YouScribe Tout savoir sur nos offres
2
pages
Français
Documents
2015
Le téléchargement nécessite un accès à la bibliothèque YouScribe Tout savoir sur nos offres
❈❤❛✐♣rt❡✶✿s✐✉❡ts
✕
rt❡♠✐♥❛❧❡✲❙♦❜❥❝❡❢✐t❇❈❆
▼❚❆❍❙✲▲Y❈❊❊✳❋❘
▼❆❚❍❙✲▲Y❈❊❊✳❋❘
❚❙✲r❡①❡❡❝✐❝r♦❝❈é❣✐✿♣✐❤❛❡✶tr
❈❤❛♣✐tr❡✶✿s♦♠♠❡❞❡st❡r♠❡s❞✬✉♥❡s✉✐t❡❡♥✉t✐❧✐s❛♥t❡❧r❛✐s♦♥♥❡♠❡♥t♣❛rré❝✉rr❡♥❝❡
❈❤❛♣r❡✐t✶✿
✳✶
✳✷
❖♥❝♦♥s✐❡r❞è❛❧✉st✐❡
❖♥♥♦t❡
❈❝❧❛✉r❡❧
☛❧✉❙♦♦♥t✐
❊♥♣❡r♥t❛♥
❊♥♣r❡♥t❛♥
▼rt♦♥❡r♣♦r✉t♦t✉❡✐❡♥tr♥❛t✉r❡❧
☛❧♦t✉❙♦✐♥
❊X❊❘❈■❈❊ ✶✼✲✷✲
❙✉✐t❡s
❡t♠♣sts❡✐♠✿é✶✵✲✶✷♠♥
2n−1
(un)❧❡r✉t❛♥r❞✜ért♦✉❡♣♥✐✐❡♥tt❡♦✉n♣r❛un=✳
n
2
n
Sn= Σuk=u0+u1+u2....+un
k=0
WWW✳▼❆❚❍❙✲▲Y❈❊❊✳❋❘
u0❡tu1✳
✿
u0=−1t❡
✿
n= 0✱ ♦♥ ❛
n= 1✱ ♦♥ ❛
1
u1=
2
−1
u0= =−1
0
2
2×1−1 1
u1= =
1
2 2
n✱ ♦♥ ❛
2n+ 3
Sn= 2−
n
2
2n+ 3
❖♥♥t❡♦Pn♣r❧❛♦étr✐♣Sn= 2−
n
2
■✲❛✐t✐♥❧✐s❛t✐♦♥
2W×0W+ 3W✳▼❆❚❍❙✲▲Y❈❊❊✳❋0❘
❖♥ ❛u0=−1❡tS0= 2−= 2−3 =−1✭r♣❡❛♣❧12 =✮
0
2
2n+ 3
❧❡ts❡✐①✉♥❡❡✐t♥r♥❛t❡✉r❧n❡❡❧q✉tSn= 2−n✮
✲❖♥✉s♣♣❡s♦q✐✬✉✭P❡✐❛r✈
n
2
2(n+ 1) + 3
t❡♦♥❡✈t✉♠❡rtr♦♥✮❡✐
✉q❡❧♦✬♥❛❧❛r♦sSn+1= 2−✭Pn+1❛r✈
n+1
2
2n+ 5
❝✬ts❡à✐❞r❡Sn+1= 2−
n+1
2
❙✐✉t❡s
P❛❣❡ ✶✴✷
▼❚❆❍❙✲▲Y❈❊❊✳❋❘t❡♠r✐♥❡❧❛❙
▼❆❚❍❙✲▲Y❈❊❊✳❋❘
t❡♠r✐♥❡❧❛✲❙♦❜❥❡✐❝❢t❇❆❈
✕
❈❤❛♣❡r✐t✶✿✉st✐s❡
❈❤❛♣❡rt✐✶✿✉s❡✐st
✕
t❡♠r✐♥❛❡❧✲❙♦❜❡❥t❝❢✐❇❈❆
▼❆❚❍❙✲▲Y❈❊❊✳❋❘
▼❚❆❍❙✲▲Y❈❊❊✳❋❘
❚❙❡✲❡①❝❡❝r❝✐r✐❣é♦
❈❤❛♣t✐❡r✿✶
❈❤❛♣✐tr❡✶✿
Sn+1=u0+u1+....+un+un+1
❙✐✉t❡s
=Sn+un+1
2n+ 3
= 2−+un+1
n
2
2n−1 2(n+ 1)−1 2n+ 2−1 2n+ 1
un=❞♦♥❝un+1== =
n n+1n+1n+1
2 22 2
❖♥ ❛ ❞♦♥❝ ✿
2n+ 32n+ 1
Sn+1= 2−+
n n+1
2 2
2(2n2+ 3)n+ 1
= 2−+
n n+1
2×2 2
4n2+ 6n+ 1
= 2−+
n+1n+1
2 2
2n+ 1−(4n+ 6)
= 2 +♥✐s❣❡−t♥❛❧❛❜❡❡❞❛✈r✐♦♥❛✈❡❝❞❡❢r❛❝t
n+1
2
2n+ 1−4n−6
= 2 +
n+1
2
−2n−5
= 2 +
n+1
2
WWW✳▼❆❚❍❙✲▲Y❈❊❊✳❋❘
2n+ 5
= 2−
n+1
2
❞♦❝♥❛❧♣r♦♣é✈r❛✐❡str✐t❣♥❛r✉❛❡n+ 1t✐♦sP❡❛✐✈r
n+1
❖♥❛❞♦♥❝♠r♦é♥t♣r❛ré✉❝r❡✉q❡♥❝❡
❙✐✉t❡s
2n+ 3
Sn= 2−♦✉rt♦✉t❡♥t✐❡r♥❛tr✉❧❡♣
n
2
WWW✳▼❆❚❍❙✲▲Y❈❊❊✳❋❘
P❛❣❡ ✷✴✷
▼❆❚❍❙▲✲Y❈❊❊❋✳❘♠r❡t♥✐❛❧❡❙
4n+ 6
n✳
▼❚❆❍❙✲▲Y❈❊❊✳❋❘
❡t♠r♥✐❧❛❡✲❙♦❜❡❥❢t✐❝❇❆❈
✕
❈❤❛✐♣❡rt✶✿✐✉ss❡t