I'm struggling with a probability theory question that asks me to show a series converges almost surely if and only if an expectation is finite. Does anyone know how I can solve this problem?
Let $X_{1}, X_{2}, \ldots$ be independent random variables all taking non-negative values. Prove that the series $\sum_{i=1}^{\infty} X_i$ converges almost surely if and only if
$$\sum_{i=1}^{\infty}E\left(\frac{X_i}{1 + X_i}\right) < \infty$$
I don't know if it helps, but the book I am studying from teaches the three series and two series theorems in this chapter. I don't know if there might be some way to cleverly apply that here with this expectation, but I have not been able to do so.
Let $Y_i=\frac {X_i} {1+X_i}$. It is easy to see that $\sum Y_i$ also converges almost surely. By the Three Series Theorem implies that $\sum EY_i <\infty$ ( becasue $0\leq Y_i \leq 1$ so no truncation is necessary). This proves one way.
The other way is easier. $\sum EY_i <\infty$ implies $\sum Y_i <\infty$ almost surely. But then ($X_i \to 0$ almost surely and ) $\sum X_i$ also converges almost surely.