The question

For the following function,

i) Find the largest open disc about the given point in which f is analytic.

ii) find the first 3 non-zero terms of the Taylor series of f about

iii) Find the coefficient of in the Taylor series of f about either as an explicit function of n, or give a recursive formula for as appropriate.

f(z) = ,

My attempt

i) I expanded the function as follows:

Clearly there's singularities at z = 3, -2

I draw a quick graph and noticed that the radius of convergence is |z - 1| < 2

This part I got correct. However, I get stuck for parts ii) and iii)

ii) I used partial fractions to get:

I then manipulated each fraction so I could write them in terms of known Macluarin series :

and

Thus I got this for the series:

Now, upon substituting n = 0, 1, 2 to get the first three terms, I get the following:

, ,

These are wrong, and I'm not sure why. :/

Any assistance would be appreciated.