Telescoping Series

A telescoping series is a sum where most terms cancel with their neighbors, leaving only a few boundary terms: $$\sum_{k=1}^{n} (a_k - a_{k+1}) = a_1 - a_{n+1}$$ Intuition: Imagine a telescope collapsing — each section slides into the next. In the sum $(a_1 - a_2) + (a_2 - a_3) + (a_3 - a_4) + \cdots + (a_n - a_{n+1})$, every interior term appears once with $+$ and once with $-$, so they all cancel. Only the first and last survive. Concrete example: Compute $\sum_{k=1}^{100} \frac{1}{k(k+1)}$. Use partial fractions: $\frac{1}{k(k+1)} = \frac{1}{k} - \frac{1}{k+1}$. $$\sum_{k=1}^{100} \left(\frac{1}{k} - \frac{1}{k+1}\right) = \frac{1}{1} - \frac{1}{101} = \frac{100}{101}$$ A 100-term sum reduced to a single subtraction! When to use: Anytime you can decompose each summand as $f(k) - f(k+1)$ (or $f(k+1) - f(k)$). Often paired with partial fractions. Common in computing sums of rational functions, proving series convergence, and deriving closed forms. Alternative approach: Partial fractions decomposition is the typical prerequisite step that reveals the telescoping structure.