Part of ALG-10 — Mathematical Induction & Summation

Telescoping Series

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Telescoping is the most powerful JEE summation technique. If the general term can be written as f(k)-f(k+1), the sum collapses to f(1)-f(n+1). Finding the telescoping form usually requires partial fractions for rational expressions. Common decompositions: 1/[k(k+1)]=1/k-1k+1\frac{1}{k+1}, 1/[k(k+2)]=12\frac{1}{2}(1/k-1k+2\frac{1}{k+2}). For the second type, the cancellation is "staggered" — the first TWO and last TWO terms survive. For triple products: 1/[k(k+1)(k+2)]=12\frac{1}{2}[1k(k+1\frac{1}{k(k+1})-1(k+1\frac{1}{(k+1}(k+2))], reducing to a single-layer telescope. The key skill is recognizing WHEN a sum telescopes: look for terms that can be expressed as differences of consecutive values of some function.

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