Expression of the sum of a series I am unable to calculate the expression of the sum of the series $1^{3/2} + 2^{3/2} + \cdots + n^{3/2}$. Could you please help me finding the answer.
 A: Notice:


*

*You can show by the limit test, that this series diverges;

*You can proof it this way:


$$\text{S}=1^{\frac{3}{2}}+2^{\frac{3}{2}}+3^{\frac{3}{2}}+\dots=\sum_{n=1}^{\infty}n^{\frac{3}{2}}=\lim_{m\to\infty}\sum_{n=1}^{m}n^{\frac{3}{2}}=$$
$$\lim_{m\to\infty}\text{H}_m^{\left(-\frac{3}{2}\right)}=\lim_{m\to\infty}-\left(\frac{3\zeta\left(\frac{5}{2}\right)}{16\pi^2}+\zeta\left(-\frac{3}{2},m+1\right)\right)=$$
$$-\frac{3\zeta\left(\frac{5}{2}\right)}{16\pi^2}-\lim_{m\to\infty}\zeta\left(-\frac{3}{2},m+1\right)\space\space\space\space\space\space\space\space\text{when}\space m\to\infty\space\text{then}\space\text{S}\to\infty$$


*

*Or using Dirichlet regulariation:


$$\lim_{s\to0}\left[\sum_{n=1}^{\infty}n^{\frac{3}{2}-s}\right]=\zeta\left(-\frac{3}{2}\right)=-\frac{3\zeta\left(\frac{5}{2}\right)}{16\pi^2}\approx -0.0254852$$
A: $\displaystyle\sum_{k=1}^nk^a~$ is known to possess a closed form expression only for $a\in\mathbb N,$ see Faulhaber's formulas 
for more information. However, we can approximate it by $~\displaystyle\int_1^{n+\tfrac12}x^a~dx~=~\bigg[~\frac{x^{a+1}}{a+1}~\bigg]_1^{n+\tfrac12}$ 
$=~\dfrac{\Big(n+\tfrac12\Big)^{a+1}-1}{a+1}.$
