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Find the series sum ln2/2 – ln3/3 + ln4/4 – ln5/5 + ….

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Find a function (or several or all) f(x) which satisfied all these three conditions: (1) lim [x→∞] f(x)/aˣ = 0 if a ≥ e (2) f(x)/aˣ → ∞ when x→∞ if 0 < a < e (3) lim [x→∞] f(x)1/x = e.