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What is the limit

$$\lim_{n\to\infty}\frac{n^{n}}{e^{n^{3/2}}}?$$

  • 7
    Hint: the numerator is $e^{n\log n}$2012-10-18
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    You can then combine them, but that is not very helpful, since whatever I do I seem to end with an indeterminate form.2012-10-18
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    Different hint: $\left(\dfrac{n}{e^{n^{1/2}}}\right)^n$. Proving the inside part approaches $0$ is easier. And if $a_n\to0$, then $a_n^n\to0$ too.2012-10-18
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    Come to think of it, isn't this a candidate sequence for the Root Test? The Root Test will imply these terms have a finite sum, which in turn proves the terms individually approach $0$.2012-10-18

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