Infinite root

Calculus Level 3

∫ 0 1 x x x x ⋯ 5 4 3 d x = ? \int_0^1x\sqrt{x\sqrt[3]{x\sqrt[4]{x\sqrt[5]{\cdots}}}}\text{d}x=?

e 1/2 2 1 1/e

Ar Hakim by AR Hakim , 25, Indonesia

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1 solution

Parth Sankhe
Oct 12, 2018

= x 1 × x 1 2 × x 1 2 × 3 . . . . . x^{1} × x^{\frac {1}{2}}× x^{\frac {1}{2×3}}.....

= x 1 + 1 2 ! + 1 3 ! + . . . . . x^{1+\frac{1}{2!} + \frac {1}{3!} + .....}

= x e − 1 x^{e-1}

(Summation of this 1 r ! \frac {1}{r!} series equals to e e when r r starts from 0)

Integral of this would be x e e \frac {x^e}{e}

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