A telescoping sum is one where each term can be written as a difference of two parts. The second part of one term cancels with the first part of the next term, thus simplifying drastically the sum's calculation. A classic example is the sum $\sum_{k=1}^{4} \frac{1}{k(k+1)}$ This looks somewhat complicated, but we can rewrite each term as $\frac{1}{k(k+1)} = \frac{1}{k} - \frac{1}{k+1}$ Now the sum becomes $ S_4 = \left(1 - \frac{1}{2}\right) + \left(\frac{1}{2} - \frac{1}{3}\right) + \left(\frac{1}{3} - \frac{1}{4}\right) + \left(\frac{1}{4} - \frac{1}{5}\right) $ The inner terms all cancel out $S_4 = 1 - \frac{1}{5} = \frac{4}{5}$ The sum *collapses* like a spy's telescope, leaving only the ends.