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For the infinite series $\sum_{ j=0}^{\infty} a_n z^n$ the Radius of convergence is given by $$R=\lim_{n \rightarrow \infty} \left| \frac{a_n}{a_{ n+1}} \right|.$$ My question is, how to find the radius of convergence of $\sum_{ j=0}^{\infty} a_n (1-z) \cdot z^n ?$

Thank you for your help.

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  • $\begingroup$ Multiply it out, then expand. $\endgroup$ – Simply Beautiful Art Feb 8 '17 at 23:16
  • $\begingroup$ I know. Then I have two series. I don't know how to find the R.O.C. then. $\endgroup$ – user394036 Feb 8 '17 at 23:19
  • $\begingroup$ BTW if $\lim_{n\to \infty} |a_n/a_{n+1}$ exists then it is the radius of convergence. In general, use the Hadamard Radius Formula, $\endgroup$ – DanielWainfleet Feb 9 '17 at 0:21
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Hint:

$$\sum_{ j=0}^{\infty} a_n (1-z) \cdot z^n=(1-z)\sum_{ j=0}^{\infty} a_n \cdot z^n$$

The $(1-z)$ does not affect convergence.

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