Find the eigenvalue and eigenfunctions of the integral operator $Ku=\int_0^\pi k(x,y)u(y)dy$. $k( x,y) = \sum\limits_{n=1}^\infty \frac{1}{n^2} \sin\big((n+1)x\big)\sin(ny)$.
This is how I approached the problem:
To find the eigenfunctions, we solve $$Ku=\lambda u$$ Since $u\in L_2[0,\pi]$, then we can expand $u(x)$ as $u(x)=\sum_{m=0}^\infty (a_m \sin(2mx)+b_m \cos(2mx))$
Now, we have
\begin{gather}\int_0^\pi \sum_{n=1}^\infty \frac{1}{n^2}\sin\big((n+1)x\big)\sin(ny)\sum_{m=1}^\infty (a_m \sin(2my)+b_m \cos(2my))\,dy=\\=\lambda \sum_{m=1}^\infty (a_m \sin(2mx)+b_m \cos(2mx))\end{gather}
$$ \sum_{n=1}^\infty \sum_{m=1}^\infty\int_0^\pi\frac{1}{n^2}\sin((n+1)x)\sin(ny) (a_m \sin(2my)+b_m \cos(2my))\,dy=\lambda \sum_{m=1}^\infty (a_m \sin(2mx)+b_m \cos(2mx))$$
I did the trivial steps, but don't know how to move next.