MHB Calculating x in Equation (1): A Telescopic Sum

[chisigma]

From mathhelpforum.com...

$\displaystyle x=\frac{1}{\sqrt{0}+\sqrt{2}} + \frac{1}{\sqrt{2}+\sqrt{4}}+...+\frac{1}{\sqrt{2006}+\sqrt{2008}}$ (1)

... what is the value of x?...

The (1) is a telescopic sum...

$\displaystyle x=\frac{\sqrt{2}}{2} + \frac{\sqrt{4}-\sqrt{2}}{2} + \frac{\sqrt{6}-\sqrt{4}}{2}+... + \frac{\sqrt{2008}-\sqrt{2006}}{2}=\frac{\sqrt{2008}}{2}=22.405356502...$ (2)

Kind regards

$\chi$ $\sigma$

 

[SammyS]

Correct.

Of course, $\dfrac{\sqrt{2008}}{2}=\sqrt{502}$ .