S₁=1-1/3=2/3,
S₂=1-1/3+1/2-1/4=11/12,
S₃=1-1/3+1/2-1/4+1/3-1/5=1+1/2-1/4-1/5=21/20.
S[n]=3/2-1/(n+1)-1/(n+2) for n>2.
S[n]=3/2-(2n+3)/((n+1)(n+2))=n(3n+5)/(2(n+1)(n+2)) for n>2.
Note that the fraction always cancels because if n is odd 3n+5 is even and if n is even 2 divides into n.
For n<3, S₁=⅔, S₂=11/12.
However, if we put n=1 and 2 into the general equation, we get 2/3 and 11/12 so the formula is valid for all n>0.