Say $X_m$ is a normal distribution with mean $0$ and variance $2m$. Now suppose we have an alternating series $-X_1+X_2-X_3+...$ up to $X_n$. Find the distribution of this series.
My working so far: I know that the moment generating function of a series of random variables is the product of each random variable's MGF. Could this lead me to the distribution?
I know the MGF of all positive $X_i$ but what is the MGF of a negative $X_i$? Is it simply the negative of the MGF of positive $X_i$?
If I'm on the wrong track, how do I find this distribution?