Telescoping (mathematics)

In mathematics, telescoping refers to a property of certain algebraic expressions or iterated binary operations in which successive terms cancel each other after expansion. As a result, the overall expression simplifies significantly, since most intermediate terms eliminate one another, leaving only a small number of boundary terms from the original structure.

Most commonly, expressions of the form

or

are considered, where is a function defined on the set of natural numbers.

Definition

Let and let .

We say that a function has the telescoping property with respect to if one of the following holds:

Additive case

.

Then the sum

is telescoping.

Multiplicative case

If , then

.

Then the product

reduces telescopically.

Relation to finite difference operator

Let be the finite difference operator:

.

Then in the additive case:

.

This is analogous to the Newton–Leibniz formula:

.

Similarly for products:

.

Examples

Telescoping sum

Decomposition:

so for we have

.

Thus:

.

Telescoping product

with , so

.

Example with Fibonacci numbers

Let be the Fibonacci sequence:

.

Then:

.

References