Difference between revisions of "Harmonic series"

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A '''harmonic series''' is a form of the zeta function :
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A '''harmonic series''' is a form of the [[zeta function]] :
 
<math> \zeta (x) = 1+\frac{1}{2^x}+\frac{1}{3^x}+\frac{1}{4^x}+... </math>.
 
<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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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>.
Euler also realized that since every number is the multiplication of some order of primes, 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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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>
  
Riemann found that when complex numbers 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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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>.
  
  
 
== How to solve ==
 
== How to solve ==

Revision as of 11:47, 24 June 2006

A harmonic series is a form of the zeta function : $\zeta (x) = 1+\frac{1}{2^x}+\frac{1}{3^x}+\frac{1}{4^x}+...$.

When $\ x$ has a value less than or equal to one the function outputs infinity. Euler found that when $\ x=2$, the zeta function outputs $\frac{\pi^2}{6}$. Euler also realized that since every number is the multiplication of some order of primes, then the zeta function is equal to $(1+\frac{1}{2^x}+\frac{1}{4^x}+...)(1+\frac{1}{3^x}+\frac{1}{9^x}+...)...(1+\frac{1}{p^x}+\frac{1}{(p^2)^x}+...)...$

Riemann found that when complex numbers are the input to the zeta function, the resulting graph is that which aids in the finding of the exact value of $\ \pi (n)$ or the number of primes less than or equal to $\ n$.


How to solve

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