General Harmonic Numbers/Examples/Order 1/Two Thirds
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Example of General Harmonic Number
- $\harm 1 {\dfrac 2 3} = \dfrac 3 2 - \dfrac 3 2 \ln 3 + \dfrac \pi {2 \sqrt 3}$
where $\harm 1 {\dfrac 2 3}$ denotes the general harmonic number of order $1$ evaluated at $\dfrac 2 3$.
Proof
\(\ds \harm 1 x\) | \(=\) | \(\ds \harm 1 {x - 1} + \dfrac 1 {x^r}\) | Recurrence Relation for General Harmonic Numbers | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \harm 1 {\dfrac 2 3}\) | \(=\) | \(\ds \harm 1 {-\dfrac 1 3} + \dfrac 1 {\frac 2 3}\) | setting $x := \dfrac 2 3$ and $r = 1$ | ||||||||||
\(\ds \) | \(=\) | \(\ds \paren {-\dfrac 3 2 \ln 3 + \dfrac \pi {2 \sqrt 3} } + \frac 3 2\) | Example: $\harm 1 {-1 / 3}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 3 2 - \dfrac 3 2 \ln 3 + \dfrac \pi {2 \sqrt 3}\) |
$\blacksquare$