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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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    Multiply it out, then expand. – 2017-02-08
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    I know. Then I have two series. I don't know how to find the R.O.C. then. – 2017-02-08
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    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, – 2017-02-09

1 Answers 1

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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.