If X_i is a series of independent, identically distributed random variables with mean m and variance v, then X_1 + X_2 + ... + X_n is approximately a normal variable with mean nm and variance vn. Therefore

(X_1 + X_2 + ... + X_n - nm)/sqrt(vn)

is approximately a standard normal.

If you draw that graph, visually the two lines should lie right on top of each other. To see the problem you need to zoom in on the tail and blow it up, and only then will you see the issues with the convergence.