How to calculate S3,S4, and S5 and then find the sum for the telescoping series ?

##\sum_{n=2}^{oo} (1/(n+1)-1/(n+2))##

##S_1=1/(2+1)-1/(2+2)= 1/3-1/4=4/12-3/12=1/12##

##S_2=1/12 + 1/(3+1)-1/(3+2)=1/12 + 1/4-1/5=1/12 + 5/20-4/20=1/12 + 1/20=2/15##

##S_3=2/15 + 1/5-1/6=2/15 +6/30-5/30=2/15+1/30=4/30+1/30=1/6##

The limit of the sum is given when

##1/12 + Lim_{n \to oo} 1/(n+2) = 1/12##