1/2-(1/3+2/3)+(1/4+2/4+3/4)-(1/5+2/5+3/5+4/5)+...-(1/21+2/21+...+19/21+20/21)
=1/2-1+(1+1/2)-2+(2+1/2)-3+.(9+1/2)-10
=(1+2+...+9)+9/2-(1+2+.+10)
=9/2-10
=-11/2
1/2-(1/3+2/3)+(1/4+2/4+3/4)-(1/5+2/5+3/5+4/5)+...-(1/21+2/21+...+19/21+20/21)
=1/2-1+(1+1/2)-2+(2+1/2)-3+.(9+1/2)-10
=(1+2+...+9)+9/2-(1+2+.+10)
=9/2-10
=-11/2