[seqfan] The relativistic harmonic numbers
Tomasz Ordowski
tomaszordowski at gmail.com
Sun Oct 3 16:21:36 CEST 2021
Relativistic sum V(n) of the velocities 1/2, 1/3, 1/4, ..., 1/n;
in units where the speed of light c = 1.
Dear readers!
We have V(n) = tanh(Sum_{k=2..n} artanh(1/k)) = N(n) / D(n) for n > 0.
Recursive definition: V(1) = 0; V(n) = (V(n-1) + 1/n) / (V(n-1)/n + 1) for
n > 1.
Hence V(n) = N(n) / D(n) = (n N(n-1) + D(n-1)) / (N(n-1) + n D(n-1)) for n
> 1.
0/1, 1/2, 5/7, 9/11, 7/8, 10/11, 27/29, 35/37, 22/23, 27/28, 65/67, 77/79,
...
It should be noted that V(n) = A160050(n) / A226089(n-1) for n > 1.
Note that lim_{n->oo} V(n) = 1.
Best regards,
Tom
___
See A160050 - OEIS <https://oeis.org/A160050> / A226089 - OEIS
<https://oeis.org/A226089>
_______
See also A348131 - OEIS <https://oeis.org/A348131> / A348132 - OEIS
<https://oeis.org/A348132>
Here the limit at infinity is tanh(1) = 0.761594...
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