Je suis complètement perdu, pouvez-vous m'aider à faire ces 2 petits exercices s'il vous plaît.
A l'aide...
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oussamaa675
Ex53: 1) U = 1+ 1/n (n≠0) n 1 1 1 1 1 1 U - U ·= (1+ ·····) - ( 1+ ······) = 1+ ······ - 1 - ····· = ······ - ····· n+1 n n+1 n n+1 n n+1 n
1(n) - 1( n+1) n - n -1 -1 = ·················= ············ =··········· n(n+1) n(n+1) n(n+1) -1 n(n+1)>0 et -1<0 ⇔; ·········< 0 ⇔ : U - U < 0 n(n+1) n+1 n ⇔ :U < U ⇔ U est décroissante : sur :[1 ; +∞[ n+1 n n
vérifier: U = 1 + 1/1 = 1+1 =2 ; U = 1+ 1/2 = 3/2 .; U = 1+1/3 = 4/3 1 2 2 2 > 3/2 > 4/3 ⇔ U > U > U >................... 1 2 3 1 1 n² +n+1 2) U - U = (n+1 +······ ) - (n + ······· ) = ·············· > 0 n+1 n n+1 n n(n+1)
⇔ U - U > 0 ⇔ U > U ⇔ U est croissante n+1 n n+1 n sur: [1 ; +∞[
vérifier: U =1+1/1 =2 ; U =2+1/2 =5/2 ; U = 3+1/3 = 10/3 1 2 3 2< 5/2< 10/3 U < U < U <········ 1 2 3
1 votes Thanks 1
Krayz
bonsoir merci à toi mais honnêtement je n'ai pas compris le détail des calculs :( je verrai peut être mieux sur papier
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1) U = 1+ 1/n (n≠0)
n
1 1 1 1 1 1
U - U ·= (1+ ·····) - ( 1+ ······) = 1+ ······ - 1 - ····· = ······ - ·····
n+1 n n+1 n n+1 n n+1 n
1(n) - 1( n+1) n - n -1 -1
= ·················= ············ =···········
n(n+1) n(n+1) n(n+1)
-1
n(n+1)>0 et -1<0 ⇔; ·········< 0 ⇔ : U - U < 0
n(n+1) n+1 n
⇔ :U < U ⇔ U est décroissante : sur :[1 ; +∞[
n+1 n n
vérifier: U = 1 + 1/1 = 1+1 =2 ; U = 1+ 1/2 = 3/2 .; U = 1+1/3 = 4/3
1 2 2
2 > 3/2 > 4/3 ⇔ U > U > U >...................
1 2 3
1 1 n² +n+1
2) U - U = (n+1 +······ ) - (n + ······· ) = ·············· > 0
n+1 n n+1 n n(n+1)
⇔ U - U > 0 ⇔ U > U ⇔ U est croissante
n+1 n n+1 n sur: [1 ; +∞[
vérifier: U =1+1/1 =2 ; U =2+1/2 =5/2 ; U = 3+1/3 = 10/3
1 2 3
2< 5/2< 10/3 U < U < U <········
1 2 3