Does the improper integral $\int_1^\infty \frac{1}{x^{1+\frac{1}{x}}}dx$ converge? This seems interesting since for $f(x) = \frac{1}{x^{1+\frac{1}{x}}}$ we have $x^{-(1+\varepsilon)} = o(f(x))$ for any $\varepsilon > 0$, but $f(x) = o(x^{-1})$, so you cannot apply the usual comparison with p-integrals.
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Also refer https://math.stackexchange.com/questions/737278/how-to-see-this-improper-integral-diverges?noredirect=1 – Riemann Sep 27 '24 at 14:27
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The linked questions solve mine perfectly; thanks! – mixotrov Sep 27 '24 at 14:35