What is the Value of this Summation?

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The forum discussion centers on the evaluation of the summation formula \[\sum_{n=1}^{9999}\frac{1}{(\sqrt{{n+1}}+\sqrt{n})(\sqrt[4]{n+1}+\sqrt[4]{n})}\]. Participants confirm the correctness of the expression and express appreciation for the mathematical formulation. The discussion highlights the significance of understanding the behavior of the summation as \(n\) approaches large values, emphasizing its convergence properties.

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\[\sum_{n=1}^{9999}\frac{1}{(\sqrt{{n+1}}+\sqrt{n}\,\,)(\sqrt[4]{n+1}\,\,+\sqrt[4]{n}\,\,)}\]
 
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Albert said:
$\sum_{1}^{9999}\dfrac{1}{(\sqrt{{n+1}}+\sqrt{n}\,\,)(\sqrt[4]{n+1}\,\,+\sqrt[4]{n}\,\,)}$

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

$\chi$ $\sigma$
 
well done !
 

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