Difference between revisions of "Harmonic series"
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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> \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 general harmonic series, <math>\displaystyle\sum_{i=1}^{\infty}\frac{1}{ai +b}</math>, has its value depending on the value of the constants <math>a</math> and <math>b</math>. | The general harmonic series, <math>\displaystyle\sum_{i=1}^{\infty}\frac{1}{ai +b}</math>, has its value depending on the value of the constants <math>a</math> and <math>b</math>. | ||
Revision as of 20:18, 22 August 2006
There are several 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.
