General Harmonic Numbers/Examples/Order 2/Half
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Example of General Harmonic Number
- $\harm 2 {\dfrac 1 2} = 4 - 2 \map \zeta 2$
where $\harm 2 {\dfrac 1 2}$ denotes the general harmonic number of order $2$ evaluated at $\dfrac 1 2$.
Proof
\(\ds \harm r x\) | \(=\) | \(\ds \harm r {x - 1} + \dfrac 1 {x^r}\) | Recurrence Relation for General Harmonic Numbers | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \harm 2 {\dfrac 1 2}\) | \(=\) | \(\ds \harm 2 {-\dfrac 1 2} + \frac 1 {\paren {\frac 1 2}^2}\) | setting $x := \dfrac 1 2$ and $r := 2$ | ||||||||||
\(\ds \) | \(=\) | \(\ds - 2 \map \zeta 2 + 4\) | General Harmonic Number of Order $2$ at $-\dfrac 1 2$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 4 - 2 \map \zeta 2\) | rearranging |
$\blacksquare$