General Harmonic Numbers/Examples/Order 2/Minus One Half

Example of General Harmonic Number

$\harm 2 {-\dfrac 1 2} = -2 \map \zeta 2$

where:

$\harm 2 {-\dfrac 1 2}$ denotes the general harmonic number of order $2$ evaluated at $-\dfrac 1 2$

$\map \zeta 2$ denotes the Riemann zeta function evaluated at $2$.

Proof

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

\(=\)

\(\ds \paren {n - n^r} \map \zeta r\)

Sum of General Harmonic Numbers in terms of Riemann Zeta Function: Corollary

\(\ds \leadsto \ \ \)

\(\ds \harm 2 {-\dfrac 1 2}\)

\(=\)

\(\ds \paren {2 - 2^2} \map \zeta 2\)

setting $z := -\dfrac 1 2$ and $r := 2$

\(\ds \)

\(=\)

\(\ds -2 \map \zeta 2\)

$\blacksquare$