The following exercise is a part of the exercise taken from the actuarial exam organized in my country. Please kindly help me, as I got stuck and have no idea how to solve it.
Assume that $\{X_n\}$ is a sequence of IID Poissonian random variables with $\lambda =3$. Calculate $\sum_{n\geq 1} E(X_n\cdot \exp (-\frac{n}{100} \cdot X_n))$.
My work so far is as follows. If $X$ is of Poisson distribution with $\lambda=3$ then $$E(X\cdot a^X) =\sum_{n\geq 1}na^n \frac{3^n}{n!} e^{-3}= 3ae^{-3} \sum_{n\geq 1} \frac{(3a)^{n-1}}{(n-1)!}=3ae^{-3}\cdot e^{3a}=3ae^{3(a-1)}.$$ Hence the desired sum is equal to $$\sum_{n\geq 1} 3e^\frac{n}{100}e^{3\left(e^\frac{n}{100}-1\right)} = \sum_{n\geq 1} 3 \exp{\left[\frac{n}{100}+ 3\left(e^\frac{n}{100}-1\right)\right]}.$$ I have no idea how to calculate the last sum. Could somebody please give some hint?