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
<ol><li>Use partial fractions, factorial identities, logarithm laws or trigonometric identities to create a difference.</li><li>Write enough opening and closing terms to see the cancellation pattern.</li><li>Keep the uncancelled boundary terms only.</li><li>For an infinite series, first derive \(S_n\), then take the limit \(n\to\infty\).</li></ol>
A term tending to zero is necessary for convergence, but it is not enough. The partial sums must approach a finite limit.

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

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

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.

Use partial fractions, factorial identities, logarithm laws or trigonometric identities to create a difference.

Write enough opening and closing terms to see the cancellation pattern.

Keep the uncancelled boundary terms only.

For an infinite series, first derive \(S_n\), then take the limit \(n\to\infty\).

A term tending to zero is necessary for convergence, but it is not enough. The partial sums must approach a finite limit.

Mathematics

International Baccalaureate Mathematics: Analysis and Approaches Higher Level

Method of differences: the telescoping idea

Method of differences: the telescoping idea