Difference between revisions of "Harmonic series"

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A '''harmonic series''' is a form of the [[zeta function]] :
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There are several types of '''harmonic series'''.
<math> \zeta (x) = 1+\frac{1}{2^x}+\frac{1}{3^x}+\frac{1}{4^x}+... </math>.
 
  
When <math>\ x</math> has a value less than or equal to one the function outputs infinity.  [[Euler]] found that when <math>\ x=2</math>, the zeta function outputs <math>\frac{\pi^2}{6} </math>.
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The the most basic harmonic series is the infinite sum
Euler also realized that since every number is the multiplication of some order of [[prime]]s, then the zeta function is equal to <math>(1+\frac{1}{2^x}+\frac{1}{4^x}+...)(1+\frac{1}{3^x}+\frac{1}{9^x}+...)...(1+\frac{1}{p^x}+\frac{1}{(p^2)^x}+...)...</math>
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<math>\displaystyle\sum_{i=1}^{\infty}\frac{1}{i}=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots</math>
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This sum slowly approaches infinity.
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The alternating harmonic series,
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<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>.
  
Riemann found that when [[complex number]]s are the input to the zeta function, the resulting graph is that which aids in the finding of the exact value of <math>\ \pi (n)</math> or the number of primes less than or equal to <math>\ n</math>.
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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>.
  
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The [[zeta-function]] is a harmonic series when the input is one.
  
 
== How to solve ==
 
== How to solve ==

Revision as of 12:45, 28 June 2006

There are several 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 $\ln 2$.

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

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