Find a power series representation for the following:

Question

Find a power series representation for the following:

 

3x / (2 - x) , centered at 1

Answer 1

Let y = x - 1.

 

Then 3x/(2 - x) = (3y + 3)/(1 - y) and the latter is to be written as a power series centered at 0.

 

(3y + 3)/(1 - y) = -3 + 6/(1 - y)

 

Use the geometric series to get

 

-3 + (1 + y + y² + y³ + ...) = -2 + y + y² + y³ + ... = -2 + ∑_n=1,...,∞ yⁿ

 

So the original expression's power series centered at 1 is

 

3x/(2 - x) = -2 + ∑_n=1,...,∞ (x - 1)ⁿ