Question

find the taylor series of -2z + 3/(z^2 -3z +2)

Answer 1

its usually best to just run the division

Answer 2

-2z + 3/(z^2 -3z +2)



3/2 +5z/4 +9z^2/8
------------------------
2 -3z +z^2 | 3 -2z
-(3 -9z/2 +3z^2/2)
------------------
5z/2 -3z^2/2
-(5z/2 -15z^2/4 +5z^3/4)
------------------------
9z^2/4 -5z^3/4
-(9z^2/4-27z^3/8 +9z^4/8)
--------------------------
17z^3/8 - 9z^4/8


3 5 9 17

looks like double and subtract 1

if you can determine a sequence for that

Answer 3

2^1 + 1
2^2 + 1
2^3 + 1

or, the top is one mor ethan the bottom

\[f(z)=\frac{2^{n+1}+1}{2^{n+1}}z^n\]