# Telescoping series

In mathematics, a telescoping series is a series whose partial sums eventually only have a finite number of terms after cancellation. This is often done by using a form of for some expression .

## Contents

## Example 1

Derive the formula for the sum of the first counting numbers.

## Solution 1

We wish to write for some expression . This expression is as .

We then telescope the expression:

.

(Notice how the sum telescopes— contains a positive and a negative of every value of from to , so those terms cancel. We are then left with , the only terms which did not cancel.)

## Example 2

Find a general formula for , where .

## Solution 2

We wish to write for some expression . This can be easily achieved with as by simple computation.

We then telescope the expression:

.

## Problems

### Introductory

- The sum can be expressed as , where and are positive integers. What is ? (Source)

- Which of the following is equivalent to (Hint: difference of squares!)

(Source)

### Intermediate

- Let denote the value of the sum can be expressed as , where and are positive integers and is not divisible by the square of any prime. Determine . (Source)

### Olympiad

- Find the value of , where is the Riemann zeta function