Telescoping series

A telescoping sum is in mathematics a finite sum of differences, cancel at any two neighboring elements (except the first and the last ) to each other. This process is called a telescoping sum. The term is derived from the telescoping of two or more cylindrical tubes.

If a sequence is, as is

A telescoping sum. Can you write a sum as a telescopic sum, their evaluation is simplified:

A series whose partial sums are telescoping sums, called telescoping series. A telescoping series

If and only convergent if converges to a limit. The sum of the series is then equal.

The same applies to the product:

Is sort of a telescopic product.

The situation is more complicated if the telescope has three (or more ) consecutive members running ( see example).

In number theory provide telescopic sums an important tool dar.

Examples

• Finite geometric series:

• Telescopic sums are often hidden a bit and can be seen for example by partial fractions:

• Triple telescopic sum: