Determine convergence or divergence of ∞Σ(n=1) 2/[n(n+2)] by expressing Sn as a telescoping sum. If convergent, find the sum.?

Sol
2/n(n+2)=[(n+2)－n]/[n(n+2)]=1/n－1/(n+2)
A=Σ(n=1 to ∞)_2/[n(n+2)]
=lim(x->∞)_[Σ(n=1 to x)_2/[n(n+2)]
=lim(x->∞)_[Σ(n=1 to x)_1/n－1/(n+2)]
=lim(x->∞)_[Σ(n=1 to x)_1/n－Σ(n=1 to x)_1/(n+2)]
Σ(n=1 to x)_1/n-Σ(n=1 to x)_1/(n+2)
=(1/1+1/2+1/3+1`/4+…+1/x)－[1/2+1/3+1/4+…+1/x+1/(x+1)]
=1/1－1/(x+1)
=x/(x+1)
So
A==lim(x->∞)_[x/(x+1)]=1