1/2000*2001+1/2001*2002+1/2002*2003···+1/2007*2008=(1/2000-1/2001)+(1/2001-1/2002)+(1/2002-1/2003)+...+(1/2007-1/2008)=1/2000-1/2001+1/2001-1/2002+1/2002-1/2003+...+1/2007-1/2008=1/2000-1/2008=1/50...