telescoping

Telescoping is a term used to describe two or more systems that fit inside one another and can slide to extend or retract to a desired length. In everyday use it refers to devices such as optical telescopes, ship masts, flagpoles, or camera sliders, where rigid tubes or sections slide in a nested arrangement. The design typically employs precise fits, bearings or bushings, and locking mechanisms to maintain alignment and length while resisting bending or twisting. Telescoping structures are valued for compact transport, adjustable reach, and modular construction, but they can be limited by play between parts, weight, and potential loss of straightness if not properly maintained.

In mathematics, telescoping describes a property of certain series or products in which successive terms cancel when the expression is expanded. A telescoping sum has terms of the form a_k - a_{k+1}; summing from k = 1 to n yields a_1 - a_{n+1}. For example, the sum from k=1 to n of (1/k - 1/(k+1)) equals 1 - 1/(n+1), which approaches 1 as n grows. Telescoping techniques are used to evaluate series, products, or limits that would be cumbersome if expanded term by term.

The origin of the term lies in the collapsing mechanism of a telescope, which can slide sections together to produce a shorter, compact instrument, paralleling the cancellation seen in the mathematical context.