The `n`

-th *harmonic number* `H`

is defined by
_{n}

```
````H`_{n} = 1 + 1/2 + 1/3 + ... + 1/n

,

so that
Hand so on._{1}= 1, H_{2}= 3/2, H_{3}= 11/6, H_{4}= 25/12, H_{5}= 137/60, H_{6}= 49/20, H_{7}= 363/140, H_{8}= 761/280, H_{9}= 7129/2520,

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

so that`a`

the integral of_{n}=`1/x`

over the interval`(n,n+1)`

,

```
````a`_{n} < 1/n < a_{n-1}

for all `n = 1,2,3,... `

, and so
```
````a`_{1}+a_{2}+...+a_{n} < H_{n} <=
1+a_{1}+a_{2}+...+a_{n-1}

for all `n = 1,2,3,... `

. Integral calculus tells us that
```
````a`_{n} = ln (n+1) - ln n

,

and so
```
``````
ln (n+1)
< H
```_{n} <=
1 + ln n

.

for all `n = 1,2,3,... `

.

One can do better yet. Since

```
````H`_{n} - ln n > H_{n} - ln (n+1) > 0

and
```
````H`_{n+1} - ln (n+1) =
(H_{n} - ln n) + (1/(n+1) - a_{n})
< H_{n} - ln n

for all `n = 1,2,3,... `

, the sequence
```
```` H`_{n} - ln n (n = 1,2,3,...)

is positive and decreasing:
Hand so on. Hence it tends to a limit. This limit is known as the_{1}- ln 1 = 1, H_{2}- ln 2 = 0.806852..., H_{3}- ln 3 = 0.734721..., H_{4}- ln 4 = 0.697038..., H_{5}- ln 5 = 0.673895..., H_{6}- ln 6 = 0.658240..., H_{7}- ln 7 = 0.646946..., H_{8}- ln 8 = 0.638415..., H_{9}- ln 9 = 0.631743...,

`gamma`

, and it equals about `0.577215...`

. To
summarize, we have
```
````gamma < H`_{n} - ln n <=
1

for all `n = 1,2,3,... `

and the lower bound is
asymptotically tight.

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