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Can a power series or a Laurent series always be analytically continued into a domain strictly larger than its convergent annulus of finite radii? Is there a way to find its maximal domain that it can analytically continue into?

As an example, what are the answers to the above questions for the following two summations from this answer?

$$p(x)=\sum_{n=0}^\infty \frac{x^n}{e^{\sqrt n}},\quad l(x)=\sum_{n=-\infty}^\infty \frac{x^n}{e^{\sqrt{|n|}}}.$$

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