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Suppose I have an analytic function, [itex]f[/itex], with radius of convergence [itex]1[/itex]. From http://planetmath.org/encyclopedia/ZerosOfAnalyticFunctionsAreIsolated.html" [Broken] it follows that any [itex]0[/itex] of [itex]f[/itex] in [itex] \{ z : |z| < 1 \}[/itex] is isolated. But is it possible that the zeroes have a limit point on the boundary of convergence? For example, is possible that an analytic function with radius of convergence 1 has zeros exactly at [itex]1-\frac{1}{n}[/itex] for all [itex]n \in Z^{+}[/itex]?

[Edit: Added question marks.]

[Edit: Added question marks.]

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