Can't get PDE solution in textbook

Hey guys having trouble getting the same answer in the text book.

Question: Consider solving wave equation (c=1) for string length Pi with fixed endpoints u(0,t)=0 and u(pi,t)=0 and initial velocity = 0 and displacement = 0.02sinx.

So after doing the usual steps i arrived at $\displaystyle u_n(x,t) = \sum_{n = 1}^\infty (Ancos(nt) + Bnsin(nt))*sin(nx)$

Using the initial conditions and fourier coefficient formula's i have Bn=0

and $\displaystyle An=\frac{2}{\pi } \int 0.02sin(x)*sin(nx) dx$

Can i treat these both as sin(x)? and use the trig indentity 1/2 -1/2cos(2x) to solve?

because the answer is u(x,t)=0.02costsinx and if i do that it will work.

but i just dont understand where the 'n' went in my solution :

$\displaystyle u_n(x,t) = \sum_{n = 1}^\infty (A_ncos(nt) + B_nsin(nt))*sin(nx)$

seeing that it should be an infinite series and the answer given is not!