Special case of Faulhaber's formula

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Consider the following sum $$S_p(n) = \sum_{k=1}^n k^p$$ for the special case $p = n$. Is there any known closed form formula/bound(s) for this case?

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By Riemann sums:

$$ \sum_{k=1}^{n} k^n = n^{n+1}\cdot\left(\frac{1}{n}\sum_{k=1}^{n}\left(\frac{k}{n}\right)^n\right)\approx n^{n+1}\int_{0}^{1}x^n\,dx = \frac{n^{n+1}}{n+1}.$$