1/x+10+1/(x+1)(x+2)+...+1/(x+9)(x+10)=3/51/(x+10)+[1/(x+1)-1/(x+2)]+...+[1/(x+9)-1/(x+10)]=3/51/(x+10)+1/(x+1)-1/(x+10)=3/51/(x+1)=3/53(x+1)=53x+3=53x=2x=2/3