# Definition:Telescoping Series

## Definition

A **telescoping series** is a series whose partial sums eventually only have a fixed number of terms after cancellation through algebraic manipulation.

## Examples

### Example 1

Let $\sequence {b_n}$ be a sequence in $\R$.

Let $\sequence {a_n}$ be a sequence whose terms are defined as:

- $a_k = b_k - b_{k + 1}$

Then:

- $\ds \sum_{k \mathop = 1}^n a_k = b_1 - b_{n + 1}$

### Example 2

Let $\left \langle {b_n} \right \rangle$ be a sequence in $\R$.

Let $\left \langle {a_n} \right \rangle$ be a sequence whose terms are defined as:

- $a_k = b_k - b_{k - 1}$

Then:

- $\displaystyle \sum_{k \mathop = m}^n a_k = b_n - b_{m - 1}$

## Also known as

A **telescoping series** is also known as a **telescoping sum**, particularly when its definition emphasises the summational nature of its structure.

Sometimes the word **concertina** is used in this context, but this is an even more informal usage.

The technique of preparing the terms into such a format is known as the **method of differences**.

## Linguistic Note

The term **telescoping series** arises from the obvious physical analogy with the folding up of a telescope.

## Sources

- 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**telescoping series**