Exercises
[1K9] Let \(c_ k\) be complex numbers, and \(a_ k=|c_ k|\). Note that power series \(β_{k=0}^β a_ k z^ k\) and \(β_{k=0}^β c_ k z^ k\) have the same radius of convergence \(R\).
Setting, for \(t{\gt}0\) real, \(\tilde f(t)=β_{k=0}^β a_ k t^ k\), note that this formula defines a monotonic function \(\tilde f:[0,β)β [0,β]\); show that the radius of convergence \(R\) coincides with the upper bound of \(tβ₯ 0\) such that \(\tilde f(t){\lt}β\).
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