# What defines a telescoping series?

## What defines a telescoping series?

Telescoping series is a series where all terms cancel out except for the first and last one. This makes such series easy to analyze.

## How do you identify a telescoping series?

The series is telescoping if we can cancel all of the terms in the middle (every term but the first and last). Canceling everything but the first half of the first term and the second half of the last term gives an expression for the series of partial sums.

How do you write a series as a telescoping series?

A telescoping series is a series where each term u k u_k uk can be written as u k = t k − t k + 1 u_k = t_{k} – t_{k+1} uk=tk−tk+1 for some series t k t_{k} tk.

Who discovered telescoping series?

Evangelista Torricelli
An early statement of the formula for the sum or partial sums of a telescoping series can be found in a 1644 work by Evangelista Torricelli, De dimensione parabolae.

### What makes a telescoping series converge?

This is the origin of the name telescoping series. This also means that we can determine the convergence of this series by taking the limit of the partial sums. In telescoping series be careful to not assume that successive terms will be the ones that cancel.

### What is sum series?

The n-th partial sum of a series is the sum of the first n terms. The sequence of partial sums of a series sometimes tends to a real limit. If this happens, we say that this limit is the sum of the series. If not, we say that the series has no sum. A series can have a sum only if the individual terms tend to zero.

What is telescoping in math?

In mathematics, a telescoping series is a series whose general term can be written as , i.e. the difference of two consecutive terms of a sequence . As a consequence the partial sums only consists of two terms of. after cancellation.

Are telescopic series convergent?

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