[Coda]

Well, the general technique is pretty difficult. XD I studied it my junior year of college, if I recall correctly.

BUT!

Let's take a look at something:

S_n = S_n-1 + 1/(n+1) - 1/(n+2)
S_n = (S_n-2 + 1/(n-1+1) - 1/(n-1+2)) + 1/(n+1) - 1/(n+2)
S_n = S_n-2 + 1/(n) - 1/(n+1) + 1/(n+1) - 1/(n+2)
S_n = S_n-2 + 1/(n) - 1/(n+2)
S_n = (S_n-3 + 1/(n-2+1) - 1/(n-2+2)) + 1/(n) - 1/(n+2)
S_n = S_n-3 + 1/(n-1) - 1/(n+2)

See a pattern there? That might be something you can work with.