meilleurs vœux pour

la période des fêtes

3
 , , ...

( ), ( ; ), ( ; ; )...

+ = +
2
5

''  = , ''
'' ( + )  ''
( )  , 
( )  , 


( )  , + +
( ; ) ( ; ) , + =

− − 

− ;  − − 

− − 
− − 

− ;  − − 
− − =

4
 

!

: ( !  ) =
: (  )  
: (  ) + + =

+ + =

+ =
=



(  )(   ) ( ; )

(  )(   ) ( ; )



  [(( ; )   ): =  =

: ( !  ) = : (  )(   ) − =
: (  ) + +  : (  )(   ) − =
: (  ) − = : (  )(   ) − =
: (  ) = : (  )(   ) − =

5
   =     
       

( )
(  ) ( ) (  ) ( )
(  ) ( ) (  ) ( )

: + = ; :  , ; :  ;  ; : (  ) = ; : (  ) 

: (  ) = : (  )(   ) =
: (  ) 
: (  )(   ) =
: (  )(   ) = : (  )(   ) =

( ) 

( ) 

( )



   

6
  =
 =
  
=  =
 (  ) 
: − =  

: + =  = 

 

 

 
 

(  ) + = (  ) − =
(  ) + = − =
(  )  (  ) 
(  )  

 
 
(  ) (  )
(  ) (  )
( ) ( )

(   ) ( ) ( ) (   ) ( ) (  ) ( )

(   ) ( ) ( ) (   ) ( ) (  ) ( )
7
(   ) ( ) ( ) (   ) ( ) (  ) ( )
(   ) ( ) ( ) (   ) ( ) (  ) ( )

(  )

, ,.....
, ,.....

,
(  )
 ( )  ( ) ( )

( ) ( ) ( )

 ( )  ( ) ( )

 ( )  ( ) ( )

( )( ) ( ) ( )
( )( ) ( ) ( )

(  ) ( ) (  ) ( )

(  ) (  ) (  ) (  )
(  )(  ) (  ) (  )
(  ) [(  ) (  )] (  ) [(  ) (  )]
8
 , , 
+
:    : = = :    :  
!
( )
: ( !  ) ( ) ( (   ) ( )) (( ;  )( ( ) ( ))  = )
 (
:  (  ) ( ) ) ((  ;  ) ( ) ( )  )

 ( ) (  )( )
(  )  [(  ) (  )] (  )(  )

( )( )
( ) ( )
(  )(  )

( )    ( )
( )    ( )

 ( )  ( ) ( )

 ( )  ( ) ( )

(  ) ( )  ( )

(  ) + − =

(( ; ) ): + =

(  )(   ): 
+

9
 
 .....  .....  .....  .... 

(  ) (  )   

(  ):   + 

(  ):   + 

  (  ) (  )

 ( )
 
 
 

 
−
− −  −
+
(  + − ) +  +


 .....  ..... 
➢
(  ) (  )   
 

 .....  ..... 

10
 (  ) −    
+
(   − ; ) −  +


 

(  )  (  ) (  )
( ;  ) + = ( = = )
+
(   ; +)(   ; +) = − + −  = =

(  ) ( ) ( )

(  ) (  )  ( ) 

(  ) − +  −
(  ) ( + )

.

 (  ) (  ) 

+
 = 
+
 = , = = .

 =   
 =

= = ( )

(  ) = ( ) =
 

11
 ( ) 

 ( )
  (  ) ( )
(  
 ) ( ) ( + )
(  ) ( )
• ( )
•  ( ) ( + )

 ( + )
(  ) + + ++ =

 +
(  ) ( + )  +
(  ) +
−

 

+ + ++ =    =
= =

+ + ++ =     = 
= =

+ ++ =   =
= =

+ + + ..... + + = ( + )   + .....  ( + ) = ( + )
= =

12
    

  =     
       

(  ) ( )
𝑄

(  ) ( )
( ) ( )
𝑃


𝑃

(  ) ( )
.

( ) ( )
( ) ( )
(∀𝑛 ∈ ℕ∗ ) 𝑃(𝑛)
(∀𝑛 ∈ ℕ) 𝑃(𝑛)

(  ) ( )
𝑃⇔𝑄

𝑃⇒𝑄

𝑃(𝑥)
𝑃

(  ) (  )
( )
(∀𝑥 ∈ 𝐸)

 [(  ) (  )]

, ,.....

, ,.....

13
 , , 
:   
: = =
+ = :  

 : ( ( ; )  ) −  +   + 
= : (   )(   )  
− =  
: (  )  
=
: ( ;  
) (  ) ( )= ( )=
+ +

: (  ) =
: (  ) − − 
(  ) + +
: (  )(   ) −
: (  )(   ) −
(( ; ) ) + =

: (  
)(  
) +  (( ; ) ) + = +

: (  )(   )  ( ) (( ; ) ) ( + )= ( )+ ( )

(( ; ) ) ( + )= ( )+ ( )

: (  )(   ) − + =
:
:
(  )   + 

(   )( =  ) (  )   + 

(  ) (   ) (  )   + − 
(   )(   )  −  ( + )  + (( ; ) ) + = + ( = = )
:    (   )( + )
(  *
) ( ;  )   (  ) + + =

( )( ) ( ;  )  ( + )( − )( − )( + )
( )( ) (  +
)   +  +

(  )( ) ( ;  ) (   ) 
+ + + +
(  ) (  ) (  ) + −   
(  ) (  ) (  ) 
  (  )  ( ;  ) (  +  ) −  −
  ( )   ( ) 
 ( ; ;  ) + 

  


 
14
 (− )
+
 
(  
)  −  = −
(  + ) + = +  = =   

(    ; + )
−

(  
)
𝑛(𝑛 + 1)(𝑛 + 2)
1 × 2 + 2 × 3 + ⋯ ⋯ + 𝑛(𝑛 + 1) =
( ;  ) + = +   3
 
(  )    +
 
( ;  ) + = ( = = )
− 

− + − + =

(  ) + +  ( )= + + − −

(  ) ( + ) (  *
): +  +

− +  + , + + 


  
−( + ) + − =
)
:
 −   + 
= , = = .
 ( + )
; 
− = ( = = ) (  
) + 

;  +

= (   )   
+ +
(  
) +
 + = +  =

( ; ) : + +   +  + 

( + )( + )
(  
)  = = + = + 
=
+ −
−
(  )  =
=
(  ) + + 
 ( + )
(  
)  = 
=  
(∀𝑛𝜖ℕ) 26𝑛+3 + 34𝑛+2
(  ) ( + )=( + ) , , ,  
+ =   +
= +
(  )  ,  + =
(  *
)  + + 
,
(  
)  ( + )
=
+
= + +  + +
( + = + ) =
15
 𝑛+1
(∀𝑛 ∈ ℕ); ∉ℕ
𝑛+2
  ( )   ( ) 

𝑎 𝑏 𝑎≠𝑏
𝑎+𝑏√2
∉ℚ
√2+1
(∀𝑛 ∈ ℕ); √𝑛2 + 5𝑛 + 8 ∉ ℕ
𝑎∈ℕ
(∀𝑛𝜖ℕ∗ ) 33𝑛+2 + 2𝑛+4
√𝑎2 + √4𝑎2 + √16𝑎2 + 8𝑎 + 3 ∉ ℕ
(∀𝑛𝜖ℕ∗ ) 52𝑛+1 + 2𝑛+4 + 2𝑛+1
(∀𝑛𝜖ℕ∗ ) 11 𝑑𝑖𝑣𝑖𝑠𝑒 32𝑛 + 26𝑛−5
1 1 1 1
(∀𝑛𝜖ℕ∗ ) (1 + 3 ) (1 + 3 ) … . (1 + 3 ) ≤ 3 − ,
1 2 𝑛 𝑛
3𝑛 1 1 1 1
(∀𝑛𝜖ℕ∗ ) ≤ 1 + 2 + 2 + ⋯ ⋯ + ≤2−
2𝑛+1 2 3 𝑛2 𝑛 𝑎+𝑏 𝑎−𝑏
𝑛 𝑘 2 𝑛(𝑛+1) | |+| | < 𝑐 ⇒ (|𝑏| < 𝑐 𝑒𝑡 |𝑎| < 𝑐)
(∀𝑛𝜖ℕ∗ ) ∑𝑘=1 = 2(2𝑛+1) 2 2
(2𝑘−1)(2𝑘+1)

(∀𝑛𝜖ℕ∗ )  ( =
+ )= ( + ) (|𝑏| < 𝑐 𝑒𝑡 |𝑎| < 𝑐) ⇒ |
𝑎+𝑏
2
|+|
𝑎−𝑏
2
|<𝑐

(− ) ( + )−
(∀𝑛𝜖ℕ∗ ) (− )
=
= , ,  (    − ; ) + + 

( + )( + )( + )  + +  − + 
(∀𝑛𝜖ℕ∗ ) ( =
+ ) =
+  
+ + 

(  
) ( ( ; ) ): ( + ) = + ,
 + 
+ + .   + 
 + 
( + )
, ,  
 + 
,  , (  
) − = ( − )  − −

= ,
,  
+ (  ) +
+ +
 + + + =  + + 


,  +

+  + 
(  )  
      ( ) / /( )
+ =
 
 (  ) +  − ( ) / /( ) =

  
(  )  +  + ( + ) , ,......,  +
  

(  
)   
  
 
( )
 = =
 
+ ( , ,  +
) ( + + ) 
,   
 + 
+

(  
) 

 
 =
−
16