I am working on a question and require some assistance. The question is:
Let $c_0, c_n$ and $d_n$ be denoted as the following: $$c_0 = \frac{1}{2}\int_{-1}^1 e^x \> dx, \>\>\>\> c_n = \int_{-1}^1 e^x\cos(n\pi x) dx, \>\>\>\>\>\> d_n = \int_{-1}^1e^x\sin(n\pi x) dx$$ What is: $$2c_0^2+\sum_{n=1}c_n^2+d_n^2$$
Now I know for the Full Fourier Series of $e^x$ these are the coefficients given. But if the coefficients are squared, it does not mean that the above will give the Fourier Series of $e^{2x}$? Am I supposed to brute force integrate these, or am I missing a trick here?