Estimate a upper bound of an infinite series.

Question

Assume $a>0$ and $a_n \geq 0$. how to verify that $$\sum_{n=1}^{\infty}\frac{a_n}{(a+S_n)^{3/2}}\leq \int_0^{\infty}\frac{1}{(a+x)^{3/2}}\mathrm{d}x$$ where $S_n = a_1+a_2+\cdots+a_n$

Thanks very much

Answer 1

$$ \int_{S_{n-1}}^{S_{n}}\frac{dx}{ (a+x)^{\frac{3}{2}}} \geq \int_{S_{n-1}}^{S_n} \frac{dx}{(a+S_{n})^\frac{3}{2}} = \frac{S_n - S_{n-1}}{(a+S_n)^\frac{3}{2}} = \frac{a_n}{(a+S_n)^\frac{3}{2}}$$