EDIT:
I'm aware I could just add a compensating "$-c_1(0)$" term to get rid of the vertical offset, but that feels not in the spirit of Fourier series... and doesn't explain the mystery anyway.
Into Desmos, I plotted:
$$f(x)=x^k\\c_1(n)=\frac{1}{\pi}\int_{-\pi}^\pi f(x)\cos(nx)\,dx\\c_2(n)=\frac{1}{\pi}\int_{-\pi}^\pi f(x)\sin(nx)\,dx\\\mathcal{F}(x)=\sum_{n=0}^Nc_1(n)\cdot\cos(nx)+c_2(n)\cdot\sin(nx)$$
The $c_1$ and $c_2$ functions are coefficient-finding functions, and rely on Desmos' numerical integration.
And this Fourier series $\mathcal{F}(x)$ fits $f(x)$ very very well, with more accuracy as $N$ increases, only when $k$ is odd. When $k$ is an even integer, the series fits the curve very well still... but at a large y-offset. The series oscillates around some y-intercept, whose value as a function of $\pi$ I have not been able to experimentally determine. For reference, here are two images, the first of $k$ odd, the second of $k$ even. The green lines are the original function, and the red lines are the Fourier series.
$k=3$ and the Fourier series fits well:
$k=4$ and the series fails... bizarre vertical offset.
Many thanks for any suggestions about where this comes from, and how to correct it.
Answer for any student who may or may not view this later, with credit to @GregMartin: the first coefficient in a Fourier series, when using this method of deriving the coefficients, must be halved: $$c_1(n)=\frac{1}{\pi}\int_{-\pi}^\pi f(x)\cos(nx)\,dx,n\gt0\\c_1(0)=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)\,dx$$
This is because in the derivation of the coefficient formulae for cosine, we have $$c_1(n)=\frac{\langle f(x),\cos(nx)\rangle}{\langle\cos(nx),\cos(nx)\rangle}$$
But $$\langle\cos(0\cdot x),\cos(0\cdot x)\rangle=2\pi$$
(but is equal to $\pi$ for all other $n$). The braces denote the inner product calculated over the range $[-\pi,\pi]$.