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I encountered the following power series, and while I know a couple of ways to determine radius of convergence, I wasn't able to figure out how to evaluate the appropriate limit to get said radius. Can anyone help?

What is the radius of convergence of the power series $\sum_{n=0}^\infty\cos\left(\alpha\sqrt{1+n^2}\right)z^n,$ where $\alpha$ is any real number? What if $\alpha$ is a complex number?

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Hint: $\sqrt{1+n^2} = n + 1/(2n) + O(1/n^3)$.

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    For the complex case you still have $\cos(\alpha \sqrt{1+n^2}) = \cos(\alpha n) (1 + O(1/n^2)) + \sin(\alpha n) O(1/n)$.2015-03-27