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TD n°5 : Suites - CORRECTI= ON

Exercice 1.

14°)

· = Démonstration 1 : La démonstration proposée sera utile pour l’étude des séries.

<= span style=3D'mso-list:Ignore'>o&nb= sp; = Montrons que la suite converg= e.

Pour =

La suite est donc décroissante et minorée par 0, = donc elle converge vers une limite avec <= /span>

<= span style=3D'mso-list:Ignore'>o&nb= sp; = On montre que :

Pour =

or <= ![if !msEquation]>

soit

<= span style=3D'mso-list:Ignore'>o&nb= sp; = Montrons alors que : la sui= te tend vers 0.

donc la suite étant positive

Par récurrence immédiate : <= /span>

or puisque , on a

Et donc par théorème d’encadrement, la suite tend vers 0.

·         = Démonstration 2 bien plus rapide.

donc

Or              donc la suite converge vers 0 (par théorème d’encad= rement).

15°)

De façon évidente,

Or la suite = diverge car les deux suites extraites convergent vers des limites différentes = 1 et -1.

Donc la suite diverge.

16°)

=

or

Donc par théorème d’encadrement, la suite converge vers 1.

17°)

<= span style=3D'font-size:11.0pt;line-height:115%;font-family:"Calibri","sans-seri= f"; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:Calibri;mso-fareas= t-theme-font: minor-latin;mso-hansi-theme-font:minor-latin;mso-bidi-font-family:"Times Ne= w Roman"; mso-bidi-theme-font:minor-bidi;mso-ansi-language:FR;mso-fareast-language:EN= -US; mso-bidi-language:AR-SA'>

=

or

La suite tend donc vers +∞

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M.Cyril, M. Peschard, Mme De Lépine, M. Duffaud&n= bsp;: http://www.math93.com/gestclasse/classes/ipsa_sp= e.html

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