Convergence of moment-generating function of weighted sums of random variables

Question

This is a continuation of my earlier question. Once again, let $c_n$ be a sequence of positive real numbers such that $$\sum^{\infty}_{n=1}{c_n} = \infty, \qquad \sum^{\infty}_{n=1}{c_n^2} < \infty.$$ Let $X_n$ be a family of i.i.d random variables with $\mathbb{E}(X_n) = 0$ and $\sigma^2(X_n) = 1$ for each $n$, and define the random variable $$X = \sum^{\infty}_{n=1}{c_n X_n}.$$ Is it true that the moment-generating function $\mathbb{E}(e^{tX})$ exists and is equal to $\prod^{\infty}_{n=1}{\mathbb{E}(e^{c_n t X_n})}$? This seems to be a bit trickier than proving the corresponding question for the characteristic function.