Find the sum of an infinite series that involve a discrete random variable

Question

I need to find the sum of the following series $\sum\limits_{n=1}^\infty \frac{X_n}{e^{n}}$ , where $X_n$ be I.I.D ,$P(X_n=1)=\frac12$ or $P(X_n=-1)=\frac12$

Using the ratio test, it can be shown that this series is convergent.Because,

$\lim \left| \frac{a_{n+1}}{a_n} \right|= lim \left| \frac{{1}}{e} \right|$<1 .

But is there any method to find the actual sum ??

can i split this sum as $\sum\limits_{n=1}^\infty \frac{1}{e^{n}}$ - $\sum\limits_{n=1}^\infty \frac{1}{e^{n}}$ since $X_n$ can only take 1 and -1 ?

Answer 1

You are only given the distribution of $X_n$'s and not a specific definition as functions on a sample space. You cannot write down the sum of the series. But one cans show that the series converges with probability $1$.