Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2008 / Paper 1 / 15 / b / i / Solution

The density is the symmetric sumFor small $\sigma$, the normal distributions are narrow and well separated, giving peaks near each nonzero integer. The largest two peaks lie near $-1$ and $1$; those near $\pm 2$ have approximately one quarter of their height, and the envelope falls as $1/n^2$. There is no component centered at zero, and the density there is exponentially small. The left panel shows the resulting smooth comb.