# 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.