= Solution
Squaring the finite geometric sum in the <Fejér kernel> gives its <Fourier series>
$$
F_n(t)=\frac1{2\pi n}\left|\sum_{j=0}^{n-1}e^{ijt}\right|^2=\frac1{2\pi}\sum_{|r|<n}\left(1-\frac{|r|}{n}\right)e^{irt}.
$$
For a fixed positive integer $k$ and $n>k$, convolution therefore gives $\sigma_n(e^{ikx})=(1-k/n)e^{ikx}$. Taking the real part,
$$
\boxed{\|\sigma_n(\cos kx)-\cos kx\|_\infty=\frac{k}{n}.}
$$
The ratio to $1/n$ is the nonzero constant $k$, even though the function is infinitely differentiable. This <Fejér saturation on a Fourier mode> disproves an $o(n^{-1})$ estimate for all $C^2$ periodic functions.
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