Find and state the convergence properties of the Taylor series for the following: f ( z ) = z 3 sin ⁡ 3 z around z 0 = 0 f ( z ) = z ( 1 − z ) 2 around z 0 = 0

Question

Find and state the convergence properties of the Taylor series for the following:  f  (  z  )  =      z    3    sin  ⁡  3  z  around       z    0    =  0  f  (  z  )  =      z          (      1      −      z              )        2              around       z    0    =  0

Gilbert Petty · Accepted Answer

The powers step inside:       z    3    sin  ⁡  3  z  =      z    3        ∑          n      =      0        ∞              (      −      1              )        n                    (      2      n      +      1      )      !            3          2      n      +      1            z          2      n      +      1        =      ∑          n      =      0        ∞              (      −      1              )        n                    3                  2          n          +          1                            (      2      n      +      1      )      !            z          2      n      +      4      As for the other one, leave the numerator   z alone for a moment, and consider       1          (      1      −      z              )        2             this is almost the derivative of a well known power series. Once you figure it out, all you have to do is multiply by the   z from the beginning.

Find and state the convergence properties of the Taylor series for the following: f(z)=z^3sin3z around z_0=0 f(z)=z/(1−z)^2 around z_0=0

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