- #1
Benny
- 584
- 0
Homework Statement
Find the radius of convergence of the following series.
[tex]
\sum\limits_{k = 1}^\infty {2^k z^{k!} }
[/tex]
Homework 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]
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.