I am aware that there are certain cases where the infinite sum does not equal the infinite integral. However, I am not yet advanced enough to be able to understand the Euler–Maclaurin formula, especially in terms of coming up with the $k$th Bernoulli Number. That said, I am wondering if there is a way somebody could explain to me the following inequality that came up in my course. When I use a calculator to estimate the infinite sum $\sum_{n=2}^{\infty}{1\over{\sqrt n(\sqrt n-1)^2}}$, I get $$\sum_{n=2}^{9999999}{1\over{\sqrt n(\sqrt n-1)^2}} \approx 7.47356$$
When I try to estimate using an integral, I get a huge discrepancy $$\int_2^\infty{1\over{\sqrt x(\sqrt x-1)^2}}dx=2^{3\over2}+2 \approx 4.8284$$
Does this require the Euler–Maclaurin formula to explain, or is there an easier way to understand what is going on over here?