Overview

Telescoping replaces many terms with a few boundary terms. The key is rewriting expressions to expose cancellation.

Key Ideas

• Use partial fractions to split rational terms.
• For products, factor each term and cancel across consecutive indices.
• Always write a few terms to see the pattern.

Core Skills

Rewrite the General Term

Look for algebraic decompositions like $\frac{1}{k(k+1)}=\frac{1}{k}-\frac{1}{k+1}$ or $\frac{1}{k^2-1}=\frac{1}{2}\left(\frac{1}{k-1}-\frac{1}{k+1}\right)$.

Track Boundary Terms

After cancellation, keep the first few and last few terms explicitly to avoid dropping a sign or a leftover term.

Products Telescope Too

Write products as factorial-like ratios so consecutive terms cancel.

Worked Example

Compute $\sum_{k=1}^{n} \frac{1}{k(k+1)}$.

Partial fractions give $\frac{1}{k(k+1)} = \frac{1}{k} - \frac{1}{k+1}$. The sum becomes $\left(1-\frac{1}{2}\right) + \left(\frac{1}{2}-\frac{1}{3}\right) + \cdots + \left(\frac{1}{n} - \frac{1}{n+1}\right)$, so everything cancels to $1 - \frac{1}{n+1}$.

More Examples

Example 1: Shifted Denominators

Compute $\sum_{k=2}^{n} \frac{1}{k^2-1}$.

Use $\frac{1}{k^2-1}=\frac{1}{2}\left(\frac{1}{k-1}-\frac{1}{k+1}\right)$. The sum telescopes to $\frac{3}{4}-\frac{1}{2(n+1)}$.

Example 2: Product

Compute $\prod_{k=2}^{n} \frac{k-1}{k}$.

Everything cancels to $1/n$.

Strategy Checklist

• Expand a few terms before committing to a pattern.
• Use partial fractions for rational sums.
• Keep boundary terms explicitly after cancellation.
• For products, factor into consecutive ratios.

Common Pitfalls

• Forgetting boundary terms after cancellation.
• Expanding products before factoring.
• Cancelling terms that are not consecutive in index.

Practice Problems