Question

How do you find a power series representation for `f(z)=z^2` and what is the radius of convergence?

Answer 1

`f(z)` is effectively a power series already with a radius of convergence of `oo`.

Explanation:

A power series (centered at `0`) is just a sum of the form `f(z) = sum_(n=0)^(oo)a_nz^n`
so in this case, `f(z)=z^2` is already a power series with
`a_n = {(1 if n = 2), (0 if n !=2):}`

In general, no manipulation is needed to find the power series of a polynomial function, as a power series is itself essentially a polynomial of infinite degree.

As for the radius of convergence, for any real value, the above power series has a single nonzero term which is equal to the square of that value, and thus does not diverge. This means the radius of convergence is infinite.