I'm investigating the asymptotic behaviour of this power series: $$f(x)=\sum_{n=1}^\infty \frac{x^n}{n^n}$$ I messed around with it in Desmos and found that $\ln(f(x))-\frac xe-\frac 12\ln(x)$ appears to converge to a constant as $x\to\infty$, implying that $f(x)=O(\sqrt xe^\frac xe)$. Reassuringly, $\frac{f(x)}{\sqrt xe^\frac xe}$ also appears to converge. $$$$ Are these results correct? If they are, is there a rigorous way to prove them? Is $\lim_{x\to\infty}\left(\frac{f(x)}{\sqrt xe^\frac xe}\right)$ a known value? Any help would be appreciated, thanks :)
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$\begingroup$ Check this: Asymptotics of the sum $\sum_{n=1}^\infty \frac{x^n}{n^n}$. $\endgroup$– Martin RCommented Sep 17, 2021 at 17:13
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$\begingroup$ And this: What's the sum of $\sum \limits_{k=1}^{\infty}\frac{t^{k}}{k^{k}}$?. $\endgroup$– Martin RCommented Sep 17, 2021 at 17:18
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