We Need To Go Radically Deeper

Algebra Level 3

$\large \sqrt{ x \sqrt[3]{x \sqrt[4]{x \sqrt[5]{ x \cdots }} }}$

Let $e = 2 + \dfrac1{1\times2} + \dfrac1{1\times2\times3}+ \dfrac1{1\times2\times3\times4} + \cdots$ .

Which of the following is equal to the expression above. (Take $x\geqslant 0$ )

x^{e-2}

x^{e-1}

x^{e+1}

x^{e+2}

1 solution

Akshat Sharda
Mar 17, 2016

$\sqrt{ x \sqrt[3]{x \sqrt[4]{x \sqrt[5]{ x \cdots }} }} = \left( x \left( x \left( x \left( x \ldots \right)^\frac{1}{5} \right)^\frac{1}{4} \right)^\frac{1}{3} \right)^\frac{1}{2} \\ x^{\frac{1}{2}}x^{\frac{1}{2×3}} x^{\frac{1}{2×3×4}}x^{\frac{1}{2×3×4×5}}\ldots= x^{\frac{1}{2}+\frac{1}{2×3}+\frac{1}{2×3×4}+\frac{1}{2×3×4×5}+\ldots} \\ \Rightarrow \boxed{x^{e-2}}$

We Need To Go Radically Deeper

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