that large integer value was first computed here: Boas, R. P. and Wrench, J. W. "Partial Sums of the Harmonic Series." Amer. Math. Monthly 78, 864-870, 1971. this on-line article has that reference, and other interesting info: http://mathworld.wolfram.com/HarmonicSeries.html bob baillie ----- Daniel Asimov wrote:
I've long wondered what is the smallest n for which
(*) 1 + 1/2 + 1/3 + . . . + 1/n > 100.
Roughly, I thought it was where ln(n) = 100, i.e.,
n ~ e^100 ~ 2.688 x 10^43.
But more exactly, it would be closer to where
100 - ln(n) = gamma (Euler's), i.e. let K = floor( e^(100-gamma) ).
Then n should be fairly close to K (which is roughly only 1.509 x 10^43).
Mathematica gives K as exactly
K = 15092688622113788323693563264538101449859497
and to my surprise it claims that
1 + 1/2 + 1/3 + . . . + 1/K > 100 but 1 + 1/2 + 1/3 + . . . + 1/(K-1) < 100 (!).
Can someone confirm that this K is exactly the least n for which (*) holds?
Thanks,
Dan
P.S. Anyone know a useful upper bound U(n) for the error
E(n) = |(1+1/2+1/3+...+1/n) - ln(n) - gamma| ? Evidently E(n) gets very small very fast.
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