Infinite Nest #1

Calculus Level 3

If x x is a positive integer, evaluate the following:

x x 2 x 3 x 4 ⋯ 6 5 4 3 \large\sqrt{x\sqrt[3]{x^{2}\sqrt[4]{x^{3}\sqrt[5]{x^{4}\sqrt[6]{\cdots}}}}}

x e x^{e} x \sqrt{x} x x x x − 1 x^{x-1}

Aidan Poor by Aidan Poor , 18, USA

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1 solution

Parth Sankhe
Oct 15, 2018

= x 1 2 + 2 3 × 2 + 3 4 × 3 × 2 . . . . x^{\frac {1}{2} + \frac {2}{3×2} + \frac {3}{4×3×2}}....

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

= x ∑ r = 1 ∞ r ( r + 1 ) ! x^{\sum_{r=1}^\infty \frac {r}{(r+1)!}}

r r + 1 ! = ( r + 1 ) − 1 r + 1 ! = 1 r ! − 1 r + 1 ! \frac {r}{r+1!} = \frac {(r+1)-1}{r+1!}= \frac {1}{r!}-\frac {1}{r+1!}

Therefore, each term except the first cancels out in this infinite summation, leaving the answer as just x 1 = x x^1=x

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