Do a long hand division of 1/{1-x}  

x greater than or equal to 1 does not result in convergence of this sum.  However this algorithm can still be used to do some interesting things.  Let us use a complex value of   x = .707+.707i  

Each power of x yields a result one step around this unit circle. Thus this series is the Z transform of the associated sequence.  [1,0] , [0.707,0.707] , [0,1] ……. This sequence is  Sin( (n-1)*pi/{4})  

Thus the z transform of this sequence is:   1/{1-(0.707+0.707i)*x}    

If you want to get express in terms of n instead of n-1 you can multiply by 1/x.  Since x is the place holder it is easy to see if you want to slide a series one unit to the left by dividing by x. 

 Sin( n*pi/{4})    : note this series starts at 45 degrees phase!

 

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