# Difference between revisions of "Harmonic series"

m (→Harmonic Series) |
|||

Line 9: | Line 9: | ||

The alternating harmonic series, | The alternating harmonic series, | ||

<math>\displaystyle\sum_{i=1}^{\infty}\frac{(-1)^{i+1}}{i}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots</math> , though, approaches <math> \displaystyle \ln 2</math>. | <math>\displaystyle\sum_{i=1}^{\infty}\frac{(-1)^{i+1}}{i}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots</math> , though, approaches <math> \displaystyle \ln 2</math>. | ||

− | |||

− | |||

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

Line 16: | Line 14: | ||

== How to solve == | == How to solve == | ||

− | ===Harmonic Series=== | + | === Harmonic Series === |

It can be shown that the harmonic series converges 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 | It can be shown that the harmonic series converges 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> | ||

− | ===Alternating Harmonic Series=== | + | === Alternating Harmonic Series === |

+ | |||

+ | === General Harmonic Series === | ||

+ | <math>\displaystyle\sum_{i=1}^{\infty}\frac{1}{ai +b}</math> is the general harmonic series, where each term is the reciprocal of a term in an arithmetic series. | ||

+ | |||

+ | '''Case 1:''' <math>a\ge b</math> | ||

+ | |||

+ | <math>ai+a\ge ai+b</math> | ||

+ | |||

+ | <math>\frac{1}{ai+b}\ge\frac{1}{ai+a}=\frac{1}{a}\left(\frac{1}{i+1}\right)</math> | ||

+ | |||

+ | <math>\sum_{i=1}^{\infty}\frac{1}{ai+b}\ge\frac{1}{a} \left(\sum_{i=1}^{\infty}\frac{1}{i+1}\right)\to\infty</math> | ||

+ | |||

+ | '''Case 2:''' <math>a<b</math> | ||

+ | |||

+ | <math>ai+b<bi+b</math> | ||

+ | |||

+ | <math>\frac{1}{ai+b}>\frac{1}{bi+b}=\frac{1}{b}\left(\frac{1}{i+1}\right)</math> | ||

+ | |||

+ | <math>\sum_{i=1}^{\infty}\frac{1}{ai+b}\ge\frac{1}{b} \left(\sum_{i=1}^{\infty}\frac{1}{i+1}\right)\to\infty</math> | ||

+ | |||

+ | Thus, <math>\sum_{i=1}^{\infty}\frac{1}{ai+b}=\infty</math> |

## Revision as of 14:13, 12 November 2006

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 converges 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,