1/1*3+1/3*5+1/5*7+1/7*9+...+1/2009*2011
=(1/2)(1-1/3+1/3-1/5+...+1/2007-1/2009+1/2009-1/2011)
=(1/2)(1-1/2011)
=(1/2)(2010/2011)
=1005/2011