I don't know if my answer is correct, can you confirm? How to prove?

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I'm worried because it seems like this is a solution that puts my thoughts on too much. Do you think this is correct?

How to prove this?

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Hint: The sum converges. Note that $$ \frac{3^n + n}{n!} \leq \frac{3^n + 3^n}{n!} $$ and apply the ratio test.