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  1. If radius of convergence of $\sum c_{n}x^{n}$ and $\sum d_{n}x^{n}$ is given say $a_{1}$ and $a_{2}$ respectively,then how can I find the radius of convergence of $\sum (c_{n}+d_{n})x^{n}$

  2. Given power series say $\sum_{n=0}^{\infty} a_{n}x^{n}$ has radius of convergence $R$,then what is the radius of convergence of $\sum a_{n}^{m}x^{n}$,$m$ is positive integer.

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closed as off-topic by Matthew Conroy, Misha Lavrov, Namaste, Chris Godsil, Shailesh Nov 28 '17 at 2:30

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  • $\begingroup$ math.stackexchange.com/questions/309466/… This post partially answers your first question. $\endgroup$ – lizardknight Nov 27 '17 at 19:44
  • $\begingroup$ You can lose the privilege of asking questions, omkar, if you continue posting low quality questions. So please take suggestions in comments and improve your posts. This is not a "do my work for me" site. $\endgroup$ – Namaste Nov 28 '17 at 1:10
  • $\begingroup$ Ask one question only, per post, $\endgroup$ – Namaste Nov 28 '17 at 1:11
  • $\begingroup$ Sorry..I will keep this in mind for future..but seriously, I have lots of respect for this site and I have never use for such homework purpose.. $\endgroup$ – Believer Nov 28 '17 at 18:45
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**just a hint **

Let $a=\min (a_1,a_2). $

if $|x|<a $ then $\sum (c_n+d_n)x^n $ converges .

for the second, the ratio $\frac {a_n^m}{a_{n+1}^m} $ goes to $R^m $ if $R $ is the radius of $\sum a_nx^n $.

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