AP Calculus BC glossary

Telescoping series

A telescoping series is one where consecutive terms cancel in the partial sum, leaving only a few surviving pieces at the ends. It is one of the rare series whose exact sum you can compute, by taking the limit of that simplified partial sum.

∑n=1∞(1n−1n+1)=1\sum_{n=1}^{\infty}\left(\frac{1}{n} - \frac{1}{n+1}\right) = 1

Partial fractions is what usually reveals the structure. Splitting one over n times n plus one into the difference of two fractions makes the cancellation visible.

The mistake

Cancelling terms without writing the partial sum. Work with the finite sum first, then take the limit; cancelling inside an infinite sum directly is not justified.

Appears in: Unit 10: Infinite Sequences and Series (BC)