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.
OK I worked out what I did wrong with part ii). I thought the series was , but it's . So my solution is correct if I just add a negative to the start of my series.
However, I'm still not sure what I'm supposed to do for part iii). Thanks.