A real periodic signal with period $T_0=2$ has the Fourier coefficients $$X_k=\left [2/3, \ 1/3e^{j\pi/4}, \ 1/3e^{-i\pi/3}, \ 1/4e^{j\pi/12}, \ e^{-j\pi/8}\right ]$$ for $k=0,1,2,3,4$. I want to calculate $\int_0^{T_0}x^2(t)\, dt$.
I have done the following:
It holds that $$\frac{1}{T_0}\int_{T_0}|x(t)|^2\, dt=\sum_{k=-\infty}^{+\infty}|X_k|^2$$ right?
Then do we get $$\int_{T_0}|x(t)|^2\, dt=2\sum_{k=-\infty}^{+\infty}|X_k|^2=2\left [\left(\frac{2}{3}\right )^2+\left(\frac{1}{3}\right )^2+\left(\frac{1}{3}\right )^2+\left(\frac{1}{4}\right )^2+1\right ]$$ But the result that I get is not one of the choices. So have I done something wrong?
The Fourier coefficients of a real-valued function have the property that $X_{-k}=X_k^*$. Do your stated Fourier coefficients have this property? Use all the Fourier coefficients that you can calculate.