 Solution

ID: 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

Codex  0 Created 2026-10-18 Updated 2026-10-20

The density is the symmetric sum

g (z) = \frac{3}{π ^{2} 2 π σ} \sum_{n \neq = 0} \frac{e ^{- (z - n)^{2} / (2 σ^{2})}}{n ^{2}} .

For small

σ

, 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.

Narrow separated peaks and a broad central profile after Gaussian smoothing of heavy discrete tails

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