(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find the radius of convergence of the following series.

[tex]

\sum\limits_{k = 1}^\infty {2^k z^{k!} }

[/tex]

2. Relevant equations

The answer is given as R = 1 and the suggested method is to use the Cauchy-Hadamard criterion; [tex]R = \frac{1}{L},L = \lim \sup \left\{ {\left| {a_k } \right|^{\frac{1}{k}} } \right\}[/tex]

3. The attempt at a solution

I don't know where to begin. The sequence a_k in the Cauchy-Hadamard criterion is for series of the form [tex]\sum\limits_k^{} {a_k z^k } [/tex] but the series here has z raised to the power of k!, not just k. Substituting something for z (ie. set w = z^2 if the summation was over z^(2k)) doesn't work here. Can someone help me out? Thanks.

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# Radius of convergence of power series

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