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Let 

\(P = 5^{1/5} \cdot 25^{1/25} \cdot 125^{1/125} \cdot 625^{1/625} \dotsm\)

 

Then P can be expressed in the form \(a^{b/c}, \) where a, b, and c are positive integers. Find the smallest possible value of a+b+c.

 May 11, 2022
 #1
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By Mathematica, P = prod(5^n^(1/5^n), 1 <= n <= inf) = 5^(4/3), so a + b + c = 5 + 4 + 3 = 12.

 May 11, 2022
 #2
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Thanks for the answer, but it's incorrect. Also, I was looking for a way perhaps using series.

 

I tried rewriting P as being equal to \({a}^{b/c} * {a}^{2b^2/c^2} * {a}^{3b^3/c^3} * ...\)

 

However, I do not know how to simplify and solve this.

Guest May 11, 2022

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