General Harmonic Number Additive Formula/Corollary

Corollary to General Harmonic Number Additive Formula

Let $n \in \N_{>0}$ be a non-zero natural number.

Then:

$\ds \sum_{k \mathop = 1}^{n - 1} \paren {\harm 1 {-\dfrac k n } } = -n \ln n$

where:

$\harm 1 x$ denotes the general harmonic number of order $1$ evaluated at $x$.

$\ln$ is the complex natural logarithm.

Proof

\(\ds \harm 1 {n x }\)

\(=\)

\(\ds \frac 1 n \sum_{k \mathop = 0}^{n - 1} \paren {\harm 1 {x - \dfrac k n } } + \ln n\)

General Harmonic Number Additive Formula

\(\ds \leadsto \ \ \)

\(\ds 0\)

\(=\)

\(\ds \frac 1 n \sum_{k \mathop = 1}^{n - 1} \paren {\harm 1 {-\dfrac k n } } + \ln n\)

$x \to 0$ and Harmonic Number $H_0 = 0$

\(\ds \leadsto \ \ \)

\(\ds \sum_{k \mathop = 1}^{n - 1} \paren {\harm 1 {-\dfrac k n } }\)

\(=\)

\(\ds - n \ln n\)

rearranging

$\blacksquare$