Summation over k to n of Harmonic Numbers over n+1-k

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Theorem

$\ds \sum_{k \mathop = 1}^n \dfrac {H_k} {n + 1 - k} = {H_{n + 1} }^2 - H_{n + 1}^{\paren 2}$

where:

$H_n$ denotes the $n$th harmonic number
$H_n^{\paren 2}$ denotes a general harmonic number.


Proof

\(\ds \sum_{k \mathop = 1}^n \dfrac {H_k} {n + 1 - k}\) \(=\) \(\ds \sum_{k \mathop = 1}^n \dfrac {H_{n + 1 - k} } k\) Permutation of Indices of Summation
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 1}^n \sum_{j \mathop = 1}^k \dfrac 1 {j \paren {n + 1 - k} }\) Definition of Harmonic Number
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 1}^n \sum_{j \mathop = k}^n \dfrac 1 {\paren {n + 1 - j} k}\) Definition of Harmonic Number


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Sources

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