I have solved the non-homogenous equation by the method of eigenfunction expansion $$u_{tt} - c^2 u_{xx}=F(x)\sin(\omega t)$$ $$0<x<L, t>0$$ $$u(x,0)=u_t(x,0)=0$$ $$u(0,t)=u(L,t)=0$$ and got the following solution: $$u(x,t)=\sum\limits_{n=1}^\infty \frac{2L\sin(\omega t)\int\limits_0^L F(x)sin\left(\frac{n\pi x}{L}\right) dx}{(cn\pi)^2-(L\omega)^2}\sin\left(\frac{n\pi x}{L}\right)$$
Now I need to find the resonant solution by taking the limit of $u(x,t)$ as $\omega\to c k\pi/L$: $$\lim\limits_{\omega\to \frac{ck\pi}{L}} u(x,t)=\sum\limits_{n=1}^\infty \frac{2L\sin(\frac{ck\pi}{L} t)\int\limits_0^L F(x)sin\left(\frac{n\pi x}{L}\right)dx}{(cn\pi)^2-({ck\pi})^2}\sin\left(\frac{n\pi x}{L}\right)$$
Can please someone let me know what is it that I'm not doing right here? Would appreciate some help.