What is the Value of this Summation?

[Albert1]

\[\sum_{n=1}^{9999}\frac{1}{(\sqrt{{n+1}}+\sqrt{n}\,\,)(\sqrt[4]{n+1}\,\,+\sqrt[4]{n}\,\,)}\]

 

[chisigma]

$\sum_{1}^{9999}\dfrac{1}{(\sqrt{{n+1}}+\sqrt{n}\,\,)(\sqrt[4]{n+1}\,\,+\sqrt[4]{n}\,\,)}$

Spoiler

Rationalizing You have...

$\displaystyle \frac{1}{(\sqrt{n+1} + \sqrt{n})\ (\sqrt[4]{n+1} + \sqrt[4]{n})} = \frac{(\sqrt{n+1} - \sqrt{n})\ (\sqrt[4] {n+1} - \sqrt[4] {n})}{\sqrt{n+1} - \sqrt{n}} = \sqrt[4] {n+1} - \sqrt[4] {n}$

... that is a 'telescopic sum'...

$\displaystyle S= \sqrt[4] {2} - \sqrt[4] {1} + \sqrt[4] {3} - \sqrt[4] {2} + ... + \sqrt[4] {10000} - \sqrt[4] {9999} = 10 - 1 = 9$

Kind regards

$\chi$ $\sigma$

 

[Albert1]

well done !