I wanna express this series $$\sqrt{1}+\sqrt{2}+\sqrt{3}+\sqrt{4}+...+\sqrt{n}$$ as Puiseux series.
According to "Wolfram Alpha", it says $$\frac{2n^{3/2}}{3}+\frac{\sqrt{n}}{2}+\zeta(-\frac{1}{2})+\frac{\sqrt{\frac{1}{n}}}{24}+O((\frac{1}{n})^1)$$
Why $\frac{2n^{3/2}}{3}$ comes out?
I just come up this question during studying Stewart's Calculus: Early Transcendentals.
Because $\frac{\mathrm d}{\mathrm dx}\frac{2x^{3/2}}3=\sqrt x$?