As stated in the title, I'm looking for a formula to express the sum $\sum_{n=a}^b n^{1.5}$ for 0 < a < b.
In order to do this, I applied the Euler-Maclaurin formula, which yields:
$$ \sum_{n=a}^b n^{1.5} = 0.4*(b^{2.5}-a^{2.5})+0.5*(b^{1.5}+a^{1.5})+\sum_{k=1}^\infty \left(\frac{B_{2k}}{(2k)!}*\prod_{i=1}^{2k-1}(2.5-i)*(b^{2.5-2k}-a^{2.5-2k})\right) $$
Note that $B_{2k}$ are the even Bernoulli numbers, and the product are the coefficients of the higher derivatives of $f(x)=x^{1.5}$, which can be factorised from the subsequent bracket.
The problem is that this entire infinite series is divergent. The summands alternate in sign, but increase in absolute value after the first few.
Now, obviously, by subtracting the regular parts of the formula from the equation, it is possible to numerically calculate the exact summation values of the infinite series for any given a and b, as it's simply the difference between the sum on the left hand side and the regular terms on the right hand side of the equation.
Therefore, shouldn't it be possible to find a general expression for the summation value of this infinite series, and hence for the sum as a whole?
I know there are summation methods to deal with infinite series, but I don't know which of them would be appropriate or opportune in this particular case and how to properly apply them. But especially with the signs in the series alternating regularly and the context (we already know that it has such a summation value, and even what that is), I reckon it should be possible to express the total sum as a function of a and b, shouldn't it?
Any help would be greatly appreciated.
EDIT: I might add, taking partial sums doesn't lead to reasonable approximations, not taking even the first summand of the series (so only the first two regular terms) yields much lower differences than taking any of them. Those differences, over a long range of increasing b, first look like a root function (which makes somewhat sense as it's the first derivative of f(x)=x^1.5 ), though with a different coefficient than the first summand has, but for higher ranges the difference behaves weird and changes smoothly, but unpredictably,leaving the stable initial root function course.