$u(x,t)=\sum_{n=1}^\infty \cos\frac{(2n-1)\pi x}{2L}(c_{1n} cos\frac{(2n-1)\pi t}{2}+c_{2n} sin\frac{(2n-1)\pi t}{2})$, $n=1,2,...$ $x\in[0,L]$
The ICs are
$u(x,0)=p(0)x$, $u_t(x,0)=p'(0)x$
Applied the first IC, I got $\sum_{n=1}^\infty c_{1n} cos\frac{(2n-1)\pi x}{2}=p(0)x$, then I don't know how to tackle with this $\sum$, anyone can please help me to walk through these two ICs?