Harmonic series

Revision as of 10:22, 23 August 2006 by JBL (talk | contribs)

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 $\displaystyle\sum_{i=1}^{\infty}\frac{1}{i}=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots$ This sum slowly approaches infinity.

The alternating harmonic series, $\displaystyle\sum_{i=1}^{\infty}\frac{(-1)^{i+1}}{i}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots$ , though, approaches $\displaystyle \ln 2$.

The general harmonic series, $\displaystyle\sum_{i=1}^{\infty}\frac{1}{ai +b}$, has its value depending on the value of the constants $a$ and $b$.

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, $\frac{1}{2}$ is greater than $\frac{1}{2}$. We also know that the third and and fourth terms, $\frac{1}{3}$ and $\frac{1}{4}$, add up to something greater than $\frac{1}{2}$. And we continue grouping the terms between powers of two. So we have $\displaystyle\sum_{i=1}^{\infty}\frac{1}{i}=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots=(1+\frac{1}{2})+(\frac{1}{3}+\frac{1}{4})+(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8})+\cdots \ge \frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\cdots \to \infty$

Alternating Harmonic Series

Invalid username
Login to AoPS