# infinite series

• December 12th 2012, 05:45 AM
muddywaters
infinite series
finding the general term for $\frac{1}{\sqrt{2}+2}+\frac{1}{2\sqrt{3}+3\sqrt{2}} +...+\frac{1}{n\sqrt{n+1}+(n+1)\sqrt{n}}+...$

$a_n=\frac{1}{n\sqrt{n+1}+(n+1)\sqrt{n}}=\frac{(n+1 )\sqrt{n}-n\sqrt{n+1}}{n(n+1)}=\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}$
how to derive $\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}$ from $\frac{(n+1)\sqrt{n}-n\sqrt{n+1}}{n(n+1)}$?
Use $\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}$ then reduce.