Sum of Harmonic Numbers approaches Harmonic Number of Product of Indices

Theorem

Let:

$\map T {m, n} := H_m + H_n - H_{m n}$

where $H_n$ denotes the $n$th harmonic number.

Then as $m$ or $n$ increases, $\map T {m, n}$ never increases, and reaches its minimum when $m$ and $n$ approach infinity.

$\ds \map T {m + 1, n} - \map T {m, n}$

$=$

$\ds \left({H_{m + 1} + H_n - H_{\paren {m + 1} n} }\right) - \paren {H_m + H_n - H_{m n} }$

$\ds $

$=$

$\ds \dfrac 1 {m + 1} + \paren {H_{\paren {m + 1} n} - H_{m n} }$

$\ds $

$=$

$\ds \dfrac 1 {m + 1} - \paren {\frac 1 {m n + 1} + \frac 1 {m n + 2} + \cdots + \frac 1 {m n + n} }$

$\ds $

$\le$

$\ds \dfrac 1 {m + 1} - \paren {\frac 1 {m n + n} + \frac 1 {m n + n} + \cdots + \frac 1 {m n + n} }$

$\ds $

$=$

$\ds \dfrac 1 {m + 1} - \frac n {n \paren {m + 1} }$

$\ds $

$=$

$\ds 0$

Similarly:

$\map T {m, n + 1} - \map T {m, n} \le 0$

$\ds \lim_{m, n \mathop \to \infty} H_m + H_n - H_{m n} = \gamma$

$\blacksquare$