Sum of General Harmonic Numbers in terms of Riemann Zeta Function/Lemma
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Lemma for Sum of General Harmonic Numbers in terms of Riemann Zeta Function
- $\ds \sum_{j \mathop = 1}^\infty \frac 1 {\paren {j + x}^r} = \sum_{k \mathop = 0}^{n - 1} \sum_{j \mathop = 1}^\infty \frac 1 {\paren {n j - k + x }^r}$
where:
- $r$ and $x$ are complex numbers
- $n \in \Z_{>0}$.
Proof
\(\ds \sum_{j \mathop = 1}^\infty \frac 1 {\paren {j + x}^r}\) | \(=\) | \(\ds \paren {\paren {\frac 1 {\paren {1 + x}^r} + \frac 1 {\paren {2 + x}^r} + \cdots + \frac 1 {\paren {\paren {n - 1} + x}^r} + \frac 1 {\paren {n + x}^r} } + \paren {\frac 1 {\paren {\paren {n + 1} + x}^r} + \cdots + \frac 1 {\paren {2n + x}^r} } + \cdots }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\paren {\frac 1 {\paren {n + x}^r} + \frac 1 {\paren {\paren {n - 1} + x}^r} + \cdots + \frac 1 {\paren {2 + x}^r} + \frac 1 {\paren {1 + x}^r} } + \paren {\frac 1 {\paren {2n + x}^r} + \cdots + \frac 1 {\paren {2 n - \paren {n - 1} + x}^r} } + \cdots }\) | reversing the order | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^{n - 1} \sum_{j \mathop = 1}^\infty \frac 1 {\paren {n j - k + x }^r}\) |
$\blacksquare$