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### Is $\sqrt1+\sqrt2+\dots+\sqrt n$ ever an integer?

Related: Can a sum of square roots be an integer? Except for the obvious cases $n=0,1$, are there any values of $n$ such that $\sum_{k=1}^n\sqrt k$ is an integer? How does one even approach such a ...
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### How closely can we estimate $\sum_{i=0}^n \sqrt{i}$

By looking at an integral and bounding the error?
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### Asymptotic behavior of the partial sums $\sum\limits_{k=1}^{n}k^{1/4}$

What is the asymptotic behavior of the sequence: $$s_n=\sum_{k=1}^{n}k^{1/4}$$ when $n\to \infty$?
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### Evaluate: $\lim_{n\to\infty}\left({2\sqrt n}-\sum_{k=1}^n\frac1{\sqrt k}\right)$

How to find $\lim\limits_{n\to\infty}\left({2\sqrt n}-\sum\limits_{k=1}^n\frac1{\sqrt k}\right)$ ? And generally does the limit of the integral of $f(x)$ minus the sum of $f(x)$ exist? How to prove ...
### Evaluating the series $\sum_{n=1}^{\infty} \frac{n}{e^{2 \pi n}-1}$ using the inverse Mellin transform
Inspired by this answer, I'm trying to show that $$\sum_{n=1}^{\infty} \frac{n}{e^{2 \pi n}-1} = \frac{1}{24} - \frac{1}{8 \pi}$$ using the inverse Mellin transform. But the answer I get is twice ...