# Logging the problem

Calculus Level 5

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

$$(\zeta +\text{log } \alpha )(\gamma \text{log } \beta )$$.

Find the value of $$\sqrt { \zeta +\alpha +\gamma +\beta }$$

• $$\zeta ,\alpha ,\gamma ,\beta$$ are all positive integers.

• $$\alpha,\beta$$ are prime numbers.

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