# Difference between revisions of "Harmonic series"

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=== Harmonic Series === | === Harmonic Series === | ||

− | It can be shown that the harmonic series | + | It can be shown that the harmonic series diverges by grouping the terms. We know that the first term, 1, added to the second term, <math>\frac{1}{2}</math> is greater than <math>\frac{1}{2}</math>. We also know that the third and and fourth terms, <math>\frac{1}{3}</math> and <math>\frac{1}{4}</math>, add up to something greater than <math>\frac{1}{2}</math>. And we continue grouping the terms between powers of two. So we have |

<math>\sum_{i=1}^{\infty}\frac{1}{i}=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots=\left(1+\frac{1}{2}\right)+\left(\frac{1}{3}+\frac{1}{4}\right)+\left(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\right)+\cdots \ge \frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\cdots \to \infty</math> | <math>\sum_{i=1}^{\infty}\frac{1}{i}=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots=\left(1+\frac{1}{2}\right)+\left(\frac{1}{3}+\frac{1}{4}\right)+\left(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\right)+\cdots \ge \frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\cdots \to \infty</math> | ||

## Revision as of 20:10, 17 March 2009

Generally, a **harmonic series** is a series whose terms involve the reciprocals of the positive integers.

There are several sub-types of **harmonic series**.

The the most basic harmonic series is the infinite sum This sum slowly approaches infinity.

The alternating harmonic series, , though, approaches .

The zeta-function is a harmonic series when the input is one.

## Contents

## How to solve

### Harmonic Series

It can be shown that the harmonic series diverges by grouping the terms. We know that the first term, 1, added to the second term, is greater than . We also know that the third and and fourth terms, and , add up to something greater than . And we continue grouping the terms between powers of two. So we have

### Alternating Harmonic Series

### General Harmonic Series

is the general harmonic series, where each term is the reciprocal of a term in an arithmetic series.

**Case 1:**

**Case 2:**

Thus,