Fourier series - time and frequency domain confusion
The coefficients of the Fourier series that you have computed are, in effect, the spectrum of the periodic signal consisting of the sum of signals $f(t)$ delayed in time or advanced in time by integer multiples of $1$ second.  Mathematically, the Fourier transform of a periodic function has impulses in it with the impulse amplitudes being the Fourier series coefficients. The impulses occur at intervals of the period, in this instance $1$ second, 
and a "plot" of the spectrum typically shows a bunch of vertical arrows (a common representation of impulses) with the amplitude marked just above the arrowhead. In your case, assuming that you have computed the complex form of the Fourier series,
you would get a series of such arrows extending from $n=0$ out to $\pm \infty$. 
Note that your formula 
$$
 \hat f_n = \frac{-(-1)^n}{2\pi ^2 n^2} + \frac{1}{2\pi ^2 n^2}
$$
is saying that $\hat{f}_n = 0$ when $n$ is even. Also, putting $n=0$ in the formula
makes $\hat{f}_0$ undefined which is incorrect since $\hat{f}_0$ is the
"DC value" which is $\frac{1}{4}$ by inspection.. So please do check your work.
In particular, note that while integrating by parts as you are likely
to be doing, the antiderivative
$$\int e^{-j2\pi nt}\,\mathrm dt = \frac{e^{-j2\pi nt}}{-j2\pi n}$$
cannot be used when $n=0$: $\int 1 \,\mathrm dt = t$.

Note: if you computed the sine/cosine form of the Fourier series, then you
should have found that the coefficients of the $\sin(2\pi nt)$ terms are all
$0$ and your spectrum will extend from $n=0$ to $n=\infty$ only.