EDB β€” 1K9

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Exercises

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

    Setting, for \(t{\gt}0\) real, \(\tilde f(t)=βˆ‘_{k=0}^∞ a_ k t^ k\), note that this formula defines a monotonic function \(\tilde f:[0,∞)β†’ [0,∞]\); show that the radius of convergence \(R\) coincides with the upper bound of \(tβ‰₯ 0\) such that \(\tilde f(t){\lt}∞\).

    Solution 1

    [1KB]

    Solution 2

    [1KC]

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  • complex numbers
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