2/(n+1)(n-1) = 1/(n-1) - 1/(n+1)

This is a "Telescoping Series" write out the partial sums.

S_2=1/1 - 1/3

S_3=1/1-1/3+1/2-1/4 = 1+1/2-1/3-1/4

S_4=1/1-1/3+1/2-1/4+1/3-1/5 = 1+1/2 - 1/4-1/5

...

It seems the general pattern is,

S_n= 1+1/2 - 1/n - 1/(n+1) for n>=2

Take n--> +oo

To find its sum.