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Assume that the radius of convergence for $\sum_{k=0}^{\infty} c_k x^k$ is $11$ and that the radius of convergence for $\sum_{k=0}^{\infty}d_k x^k$ is $13$. Determine the radius of convergence for $\sum_{k=0}^{\infty}(c_k + d_k) x^{3k}$.

I remember to have ever read something that the radius of convergence of the sum of two series is equal to the lowest r.o.c. of the two series, which would mean the answer would be $11$. However, I am not completely sure that this holds, especially since the questions ask to compute it for $x^{3k}$. Could anyone please help me out?

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  • $\begingroup$ Could you please check if the text written like now is right ? i interpreted the text different $\endgroup$ – Dominic Michaelis Feb 22 '13 at 20:18
  • $\begingroup$ I guess it is $c_k$ and $d_k$ otherwise radius of convergence would be the same. $\endgroup$ – Maesumi Feb 22 '13 at 20:26
  • $\begingroup$ not only that, if the brackets are as they are now the roc would be $\sqrt[3]{13}$ $\endgroup$ – Dominic Michaelis Feb 22 '13 at 20:30
  • $\begingroup$ I edited the question so it makes sense to me (and previous comments), please check. $\endgroup$ – vonbrand Feb 22 '13 at 22:25
  • $\begingroup$ @vonbrand yeah that one i thought too :) $\endgroup$ – Dominic Michaelis Feb 22 '13 at 22:27
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It is the lowest only if they are different, you substitute $z=x^3 $ and hence the r.o.c. is $\sqrt[3]{11}$ For the proof i will search my answer and link it.

proof

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  • $\begingroup$ Thanks a lot! Highly appreciated! $\endgroup$ – dreamer Feb 22 '13 at 20:04

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