Addition Addiction

Calculus Level 4

\[\left \lceil \displaystyle \sum_{a=1}^{5} \left ( \displaystyle \sum_{b=1}^{\infty} \displaystyle \dfrac{b^a}{b!} \right ) \right \rceil = \: \: ?\]


This question was inspired by a Harvard-MIT Mathematics Tournament Calculus question.
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