Thursday, 11 September 2008

About harmonic series.

*JSL* Harmonic series is a special sequence that touch infinitive but very slow. There's so many ways to prove it. Let harmonic series be S and it x th partial sum if S(s). S=1+1/2+... =1 + (1/2) + (1/3+1/4) +(1/5+1/6+1/7+1/8)...←→S(1)+(S(2)-S(1))+(S(4)-S(2))...+(S(2^p)-S(s^(p-1))... >1+1/2+1/2+1/2....... => infinite. *JSL+* The let of harmoninc series's notation is still exist. S=(1+1/3+1/5...)+(1/2+1/4+1/6...) =(1+1/3+1/5...)+0.5S >(1/2+1/4+1/6...)+0.5S+0.5 =S+0.5 S>S+0.5 only exist when S is infinite. *SSL* The sum of prime^-1 is also infinite. Prove. Notation: P is the sum and P(x) is partial sum. S is the sum of harmonic series. ln S = Sum of (Prime)ln 1/(1-p^-1)= sum of (prime) -(ln (1-p^-1)) ln S = infinite

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