From: Joe Keane <jgk@jgk.org>
Newsgroups: sci.math
Subject: Re: Harmonic series
Date: 3 Dec 1996 03:29:48 -0800
Message-ID: <5812vc$90b@shellx.best.com>

With a bit of thought and consideration of symmetry principles, it's not
hard to see that the approximation H_n ~= log n + C is `off by half'.

One way to fix things is use only half of the last term:

    H'_n ~ log n + C - 1/12 * n^-2 + 1/120 * n^-4 - 1/252 * n^-6 + ...

        where H'_n = H_n - 1/2 * n^-1

Another way is to shift the expansion point by one half:

    H_n ~ log m + C + 1/24 * m^-2 - 7/960 * m^-4 + 31/8064 * m^-6 - ...

        where m = n + 1/2

It's not clear what the constant C should be, but it's easy to see that
it must be the same in both series.  In fact either of these asymptotic
expansions could be taken as the definition of Euler's constant gamma,

The two series are very similar; both involve the Bernoulli numbers, and
it's not hard to guess the relationship between them.  The second series
starts a bit smaller, but asymptotically they're the same.

--
Joe Keane, amateur mathematician