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**Prove It** You probably need to start by rationalising the denominator.

$\displaystyle \displaystyle \begin{align*} \frac{1}{n\sqrt{n + 1} + (n + 1)\sqrt{n}} &= \frac{n\sqrt{n + 1} - (n + 1)\sqrt{n}}{[n\sqrt{n + 1} + (n + 1)\sqrt{n}][n\sqrt{n + 1} - (n + 1)\sqrt{n}]} \\ &= \frac{n\sqrt{n + 1} - (n + 1)\sqrt{n}}{n^2(n + 1) - n(n + 1)^2}\\ &= \frac{n\sqrt{n + 1} - (n + 1)\sqrt{n}}{n(n + 1)[n - (n + 1)]} \\ &= \frac{(n + 1)\sqrt{n} - n\sqrt{n + 1}}{n(n + 1)} \\ &= \frac{(n + 1)\sqrt{n } - n\sqrt{n + 1}}{(\sqrt{n})^2(\sqrt{n + 1})^2}\end{align*}$

Now try the Partial Fractions decomposition.