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

1 Answer
Nov 20, 2015

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


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.