= Solution
{c}
First, the <Fourier coefficients> of the <Fourier cosine series> decay as $O(n^{-2})$, whereas those of the <Fourier sine series> are of order $n^{-1}$. In particular, the <absolute convergence> bound for the <Fourier cosine series> gives a uniform tail estimate
$$
\sup_{0\leq x\leq L}|g(x)-C_N(x)|
\leq2(e+1)\sum_{n>N}\frac1{1+n^2\pi^2}=O(N^{-1}).
$$
Thus its tail is absolutely controlled using fewer <Fourier harmonics>. The <Fourier sine series> does not have an absolutely summable sequence of <Fourier coefficients>.
Second, the <even extension> is continuous, so its <Fourier cosine series> converges uniformly on the closed interval and approximates the nonzero endpoint limits correctly. The <odd periodic extension> has jumps, producing the <Gibbs phenomenon> in its <Fourier sine series> near zero and $L$. Every <Fourier sine series> <partial sum> is zero at both endpoints, so by approaching $L$ from inside $(0,L)$ one obtains
$$
\sup_{0<x<L}|g(x)-S_N(x)|\geq e.
$$
Hence the <Fourier sine series> approximation cannot converge uniformly even on the whole open interval, despite <pointwise convergence> at each interior point. \b[The <Fourier cosine series> approximation has faster <coefficient decay> and avoids the persistent errors near the endpoints.]
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