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Find all values of x that satisfy \(x=1-x+x^2-x^3+x^4-x^5+ ...\)

 #1
avatar+214 
+1

And I don't even get this question, so sad

 #2
avatar+128794 
+1

x - 1  =   x (x +x^3 + x^5 +.....)   -  1(x + x^3  + x^5 + .....)

 

x - 1 =  (x -1) (x + x^3 + x^5 + .......)

 

(x -1)   / ( x -1)  =   ( x + x^3 + x^5  +  ......)

 

1  =  ( x + x^3 + x^5 +  .......)

 

-1 =  -x - x^3 - x^5 +  .........

 

So

 

x = 1 - 1  + x^2 +x^4 + x^6+ ........

 

x = x^2 + x^4 + x^6 + .......

 

Infinite sum 

 

x^2 / ( 1 - x^2)  =  x

 

x^2  =  x - x^3

 

x^3 + x^2   - x  = 0

 

x ( x^2 + x - 1)   = 0

 

x = 0   reject

 

x^2 + x - 1   = 0

 

x^2 + x  = 1

 

x^2 + x + 1/4 =  1 + 1/4

 

(x + 1/2)^2  = 5/4

 

x + 1/2  = sqrt (5)  / 2          or    x + 1/2  =  -sqrt (5) / 2

 

x = [ sqrt (5) - 1 ]  2            or  x =  [ -sqrt (5)  -1 ] / 2

 

x =  "phi"      or    x =  "-Phi "

 

 

cool cool cool

 Apr 7, 2024
 #3
avatar+214 
+1

Aprreciate it smiley


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