General Harmonic Numbers/Examples/Order 2/Minus One Half
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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 $r := 2$ | ||||||||||
\(\ds \) | \(=\) | \(\ds -2 \map \zeta 2\) |
$\blacksquare$