EDB — 1K9

[1K9] Let \(c_ k\) be complex numbers, and \(a_ k=|c_ k|\). Note that power series \(∑_{k=0}^∞ a_ k z^ k\) and \(∑_{k=0}^∞ c_ k z^ k\) have the same radius of convergence \(R\).

Setting, for \(t{\gt}0\) real, \(\tilde f(t)=∑_{k=0}^∞ a_ k t^ k\), note that this formula defines a monotonic function \(\tilde f:[0,∞)→ [0,∞]\); show that the radius of convergence \(R\) coincides with the upper bound of \(t≥ 0\) such that \(\tilde f(t){\lt}∞\).