# Difference between revisions of "Harmonic series"

m |
m (→Harmonic Series) |
||

Line 19: | Line 19: | ||

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> | + | <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=== |

## Revision as of 10:48, 5 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 general harmonic series, , has its value depending on the value of the constants and .

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

## 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