jmc

We have that $$\begin{array}{rl}\frac{1}{n\sqrt{n-1}+(n-1)\sqrt{n}}& =\frac{n\sqrt{n-1}-(n-1)\sqrt{n}}{(n\sqrt{n-1}+(n-1)\sqrt{n})(n\sqrt{n-1}-(n-1)\sqrt{n})}\\ & =\frac{n\sqrt{n-1}-(n-1)\sqrt{n}}{n^2(n-1)-(n-1{)}^{2}n}\\ & =\frac{n\sqrt{n-1}-(n-1)\sqrt{n}}{n(n-1)(n-(n-1))}\\ & =\frac{n\sqrt{n-1}-(n-1)\sqrt{n}}{n(n-1)}\\ & =\frac{1}{\sqrt{n-1}}-\frac{1}{\sqrt{n}}.\end{array}$$Thus, $$\begin{array}{rl}\sum _{n=2}^{10000}\frac{1}{n\sqrt{n-1}+(n-1)\sqrt{n}}& =\left(1-\frac{1}{\sqrt{2}}\right)+\left(\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}\right)+\left(\frac{1}{\sqrt{3}}-\frac{1}{\sqrt{4}}\right)+\cdots +\left(\frac{1}{\sqrt{9999}}-\frac{1}{\sqrt{10000}}\right)\\ & =1-\frac{1}{100}=\boxed{\frac{99}{100}}.\end{array}$$