Telescoping Series

A telescoping series is a series where most of the terms cancel each other out, leaving only a finite number of terms. This type of series often involves partial fractions and can be simplified by expressing it as a difference of two expressions.

How It Works

In a telescoping series, each term in the series contains both positive and negative terms that can be combined to simplify the series. As you expand the series by adding more terms, these cancellations occur, leading to a convergence of the series to a finite value. The result is that the infinite sum is reduced to a finite sum, making it easier to calculate.

Example

An example of a telescoping series is the series ( sum_{n=1}^{infty} left(frac{1}{n} - frac{1}{n+1} ight) ). By expanding this series, you can see that most terms cancel out, leaving only ( 1 - frac{1}{n+1} ) as the final term. As n approaches infinity, this series converges to the value of 1.

Telescoping series are a useful tool in mathematics and are often used to demonstrate convergence of series in a simplified manner.