<s>[INST] Value of $\frac{1}{\sqrt{3}+\sqrt{7}}+\frac{1}{\sqrt{7}+\sqrt{11}}+\frac{1}{\sqrt{11}+\sqrt{15}}+\cdots$ ($n$ terms) Sum $$\frac{1}{\sqrt{3}+\sqrt{7}}+\frac{1}{\sqrt{7}+\sqrt{11}}+\frac{1}{\sqrt{11}+\sqrt{15}}+\cdots \text{ ($n$ terms)}$$
I know how to use the telescoping series method when the terms are in product. Here i tried to make $f(n+1)-f(n)$ form but failed to do so. How do we go about with this one?
Any help is welcome,
Thanks.
 [/INST] Multiplying top and bottom by the denominator with a $-$ sign, we can rewrite this as
$$\frac{\sqrt 7 - \sqrt 3}{7 - 3} + \frac{\sqrt{11} - \sqrt 7}{11 - 7} + \frac{\sqrt{15} - \sqrt{11}}{15 - 11} + \dots + \text{ final term}$$
This does telescope.
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