Method of differences: the telescoping idea

The target form

Rewrite the general term as a difference of shifted terms, for example \(u_r=f(r)-f(r+1)\). Then

\[\sum_{r=1}^{n}[f(r)-f(r+1)]=f(1)-f(n+1).\]

Why it works

The middle terms cancel when the first few and last few terms are written out. A shift of two leaves two terms at the front and two at the end.

Reliable routine

1. Use partial fractions, factorial identities, logarithm laws or trigonometric identities to create a difference.
2. Write enough opening and closing terms to see the cancellation pattern.
3. Keep the uncancelled boundary terms only.
4. For an infinite series, first derive \(S_n\), then take the limit \(n\to\infty\).

IB AA scope: this is a useful application of summation and algebraic cancellation. “Method of differences” is not a separate prescribed AA topic; any unfamiliar decomposition should be introduced or derived.