Harmonic Series

n sn

With sn=1+1/2+1/3+...+1/n, the harmonic series
limn sn= 1+ 1/2+ 1/3+ 1/4+ 1/5+ 1/6+ 1/7+ 1/8+ 1/9+ 1/10+ 1/11+ 1/12+ 1/13+ 1/14+ 1/15+ 1/16+ ...
> 1+ 1/2+ 1/4+ 1/4+ 1/8+ 1/8+ 1/8+ 1/8+ 1/16+ 1/16+ 1/16+ 1/16+ 1/16+ 1/16+ 1/16+ 1/16+ ...
= 1 + 1/2 + 1/2 + 1/2 + 1/2 + ...
diverges - but so slowely, that a numerical experiment does not show that. Even if a machine would have been adding terms at a rate of 10-9 seconds and would have started 15 billion years ago, (about 1017 seconds), the value of the sum would still be about Log(1026) which is less then 60.
A similar discrepancy between the mathematical fact and experience can be observed in the Petersburg paradox or with recurrence theorems in thermodynamics.
© Mathematik.com 2001

 

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