Logging the problem

Calculus Level 5

If an=(log 3)nr=1nr2r!(nr)!,nN\displaystyle a_{n}= (\text{log } 3)^{ n } \sum _{ r=1 }^{ n }{ \frac { { r }^{ 2 } }{ r!(n-r)! } } ,\quad n\in \mathbb{N} , then the sum of the series a1+a2+a3+{ a }_{ 1 }+{ a }_{ 2 }+{ a }_{ 3 } + \ldots can be represented as

(ζ+log α)(γlog β)(\zeta +\text{log } \alpha )(\gamma \text{log } \beta ).

Find the value of ζ+α+γ+β\sqrt { \zeta +\alpha +\gamma +\beta }


  • ζ,α,γ,β\zeta ,\alpha ,\gamma ,\beta are all positive integers.

  • α,β\alpha,\beta are prime numbers.

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