Long hand division generation of polynomials

Do a long hand division of [pmath] 1/{1-x} [/pmath] 

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  [pmath] x = .707+.707i [/pmath] 

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 [pmath] Sin( (n-1)*pi/{4}) [/pmath] 

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

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. 

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

More information:

• Generating Functions
• Z Transform