Sum of the square roots of the first n natural numbers

[Amith2006]

Is there a way to find the,"Sum of the square roots of the first n natural numbers"?

 

[StatusX]

I don't think you could do it exactly. You could approximate it by the integral of $\sqrt{x}$, and get a bound on the error.

 

[Data]

As StatusX says I'm pretty sure there's no way to do in exactly in closed-form. If you don't have a way to calculate square roots at all (ie. you're doing it without a calculator and don't want to go through an approximation method), then a simple integer approximation would be

$$\frac{2}{3}\lfloor \sqrt{n} \rfloor^3 - \frac{1}{2}\lfloor \sqrt{n} \rfloor^2 - \frac{1}{6} \lfloor \sqrt{n} \rfloor + \lfloor \sqrt{n} \rfloor(n-\lfloor \sqrt{n} \rfloor^2),$$

but it's not very good. The integral approximation $\frac{2}{3} n^{\frac{3}{2}}$ is much better, but you have to be able to compute $n^{3/2}$
($2/3 \lfloor n^{3/2}\rfloor$ is also better than the one I gave above though).