1/n*(n+1)=1/n-1/(n+1)
所以1/1*2+1/2*3+1/3*4...+1/99*100=1-1/2+1/2-1/3+1/3+1/4...+1/99-1/100=1-1/100=99/100
所以1/1*2+1/2*3+1/3*4...+1/99*100+1/(2011*2012)=99/100+1/（2011*2012）=400567168/404613200=0.99