Harmonic numbers, natural logarithms, and the Euler-Mascheroni constant

The n-th harmonic number Hn is defined by

Hn = 1 + 1/2 + 1/3 + ... + 1/n,
so that 
      H1 =     1,
      H2 =    3/2,
      H3 =   11/6,
      H4 =   25/12,
      H5 =  137/60,
      H6 =   49/20,
      H7 =  363/140,
      H8 =  761/280,
      H9 = 7129/2520,
and so on.


It turns out that these numbers are well approximated by the natural logarithm. To work out the details, let us write

an = the integral of 1/x over the interval (n,n+1),
so that
an < 1/n < an-1
for all n = 1,2,3,... , and so
a1+a2+...+an < Hn <= 1+a1+a2+...+an-1
for all n = 1,2,3,... . Integral calculus tells us that
an = ln (n+1) - ln n,
and so
ln (n+1) < Hn <= 1 + ln n.
for all n = 1,2,3,... .


One can do better yet. Since

Hn - ln n > Hn - ln (n+1) > 0
and
Hn+1 - ln (n+1) = (Hn - ln n) + (1/(n+1) - an) < Hn - ln n
for all n = 1,2,3,... , the sequence
Hn - ln n     (n = 1,2,3,...)
is positive and decreasing: 
      H1 - ln 1 =  1,
      H2 - ln 2 =  0.806852...,
      H3 - ln 3 =  0.734721...,
      H4 - ln 4 =  0.697038...,
      H5 - ln 5 =  0.673895...,
      H6 - ln 6 =  0.658240...,
      H7 - ln 7 =  0.646946...,
      H8 - ln 8 =  0.638415...,
      H9 - ln 9 =  0.631743...,
and so on. Hence it tends to a limit. This limit is known as the Euler-Mascheroni constant, (see the web page from Eric Weisstein's MathWorld ), it is usually denoted by gamma, and it equals about 0.577215... . To summarize, we have
gamma < Hn - ln n <= 1
for all n = 1,2,3,... and the lower bound is asymptotically tight.

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