Telescoping series or sum means the sum of a sequence of terms of
the form

** (a_1-a_0) + (a_2-a_1) + (a_3-a_2) + ... + (a_n - a_(n-1)
)**

simplifies to just **a_n-a_0** because if you just rearrange
all the terms and do all the obvious cancelling, this is all
that's left.

So we can sum ** 1/(k*(k+1))** for k from 1 to n, by realizing
that this is the same as
** (a_1-a_0) + (a_2-a_1) + (a_3-a_2) + ... + (a_n - a_(n-1)
)**

where we let **a_j= -(1/j)**. Why the minus sign? Well, our
sum is of the terms

** 1/(j*(j+1)) = 1/j - 1/(j+1) = (-1/(j+1) - (-1/j) = a_j -
a_(j+1) **.

It is probably simpler to write this out directly:

** 1/1*2 + 1/2*3 + ... + 1/n*(n+1) =
(1/1-1/2) + (1/2-1/3) + ... + (1/n - 1/(n+1)) **

** = (1/1) - (1/(n+1)) **
It remains to verify that ** 1/(k*(k+1)) ** is equal to **1/k
- 1/(k+1)**. Just add the fractions together by putting them
over the common denominator **k*(k+1)**.