99/100
Take the first four terms: 1/2+1/6+1/12+1/20=4/5.
In general, the first n terms sum to n/(n+1).
To prove by induction, assume S[n]=n/(n+1) then
S[n+1]=n/(n+1)+1/((n+1)(n+2))=
(n(n+2)+1)/((n+1)(n+2))=
(n²+2n+1)/((n+1)(n+2))=
(n+1)²/((n+1)(n+2))=(n+1)/(n+2)=S[n+1]
When n=1 (base case), S₁=½ and S₂=1/(1×2)+1/(2×3)=½+⅙=⅔. And we expect this using the formula.
In the question n=99 so S₉₉=99/100.