Set H=L=1

u[x_,y_]:=Sum[((20(1-(((-1))^n)))/(n*π*Sinh[n*π*(L/H)]))Sinh[((n*π)/H)x]Sin[((n*π)/H)y],{n,1,80}]/.H->1/.L->1

u[x,y]

Plot3D[u[x,y],{x,0,1},{y,0,1},AxesLabel->Automatic]

Integrate[10 Sin[n (π/H)y],{y,0,H}]