Telescoping series

Definition

A telescoping series is an infinite series where most terms cancel out with subsequent terms, leaving only a few terms to sum. This characteristic makes it easier to find the series' sum.

5 Must Know Facts For Your Next Test

1. A telescoping series typically has the form $\sum_{n=1}^{\infty} (a_n - a_{n+1})$.
2. The partial sums of a telescoping series simplify significantly due to cancellation of intermediate terms.
3. To find the sum of a telescoping series, identify the first and last remaining terms after cancellation.
4. Convergence can often be determined by examining if the remaining terms approach a limit as $n$ approaches infinity.
5. Telescoping series are useful for evaluating complex-looking sums that simplify through cancellation.

Review Questions

• What is the general form of a telescoping series?
• How do you determine the sum of a telescoping series?
• What happens to most intermediate terms in a telescoping series?

Related terms

Infinite Series: A sequence of numbers that are added together indefinitely.

Partial Sum: The sum of the first $n$ terms in an infinite series.

Convergence: A property indicating that an infinite series approaches a finite value as more terms are added.