Definition:Telescoping Series

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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 $\left \langle {b_n} \right \rangle$ 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:

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


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