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.