Hi, I’ve been trying to crack this problem but I’m unable to make any progress, could you please give me hints? Thx
Edit: The solution is from wolframalpha
Simply use
$$ \sum_{i=0}^{\infty} \frac{(an)^ie^{-an}e^{-i/n}}{i!}=e^{-an} \sum_{i=0}^{\infty} \frac{(ane^{-1/n})^i}{i!}=e^{-an} e^{ane^{-1/n}} $$