已知函数f（x）=sinπ/ 2x,则f（1）+.+f（2012)+f(2013)

当x=4kπ+1时,π/2x=2kπ+π/2 sinπ/2x=cos(2kπ+π/2)=1
当x=4kπ+2时,π/2x=2kπ+π sinπ/2x=cos(2kπ+π)=0
当x=4kπ+3时,π/2x=2kπ+3π/2 sinπ/2x=cos(2kπ+3π/2)=-1
当x=4kπ+4时,π/2x=2kπ+2π sinπ/2x=cos(2kπ+2π)=0
∴f(4k+1)+f(4k+2)+f(4k+3)+f(4k+4)=0
∴f(1)+f(2)+...+f(2009)=[f(1)+f(2)+f(3)+f(4)]+……+[f(2005)+f(2006)+f(2007)+f(2008)]+[f(2009)+f(2010)+f(2011)+f(2012)]+f(2013)=f(2013)=sin(2013π/2) =sin(1006π+π/2)=sin(π/2)=1