If two fourier series are equal, are the individual terms equal too?
I have
$$\sum_{k=1}^{\infty}\left((u_k'(t)-(\pi k)^2u_k(t))\sin(k\pi x)\right) = \sum_{k=1}^\infty c_ke^{-t}\sin(k\pi x)$$ with $c_k = 0 \ \forall k\neq 1$. Can I separate this equality into different cases ($k=1..2.....)$ and solve for $u_1, u_2$ individually? This example might be easy but I am wondering in general.
