How do I determine if a function is analytic at x=a?

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To determine if a function is analytic at a point x=a, one must verify that it has a Taylor series expansion around that point. This involves checking if the function is differentiable in a neighborhood of x=a and if the derivatives at that point exist. A common approach is to compute the function's derivatives and evaluate them at x=a to construct the Taylor series. If the series converges to the function in some interval around x=a, the function is considered analytic there. Understanding these criteria is essential for solving differential equations effectively.
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Homework Statement


I'm studying diff eq and I'm not sure how to show that a function is analytic at point x=a. I know that the definition says that a function p(x) is analytic if it has a Taylor series expansion at x=a, but not sure how to show it.

Thank you for the help!

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