Summation to n of kth Harmonic Number over k+1
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Theorem
- $\ds \sum_{k \mathop = 1}^n \dfrac {H_k} {k + 1} = \dfrac { {H_{n + 1} }^2 - \harm 2 {n + 1} } 2$
where:
- $H_n$ denotes the $n$th harmonic number
- $\harm 2 n$ denotes the general harmonic number of order $2$ evaluated at $n$.
Proof
\(\ds \sum_{k \mathop = 1}^n \dfrac {H_k} {k + 1}\) | \(=\) | \(\ds \sum_{k \mathop = 1}^n \dfrac 1 {k + 1} \paren {H_{k + 1} - \dfrac 1 {\paren {k + 1} } }\) | Definition of Harmonic Number | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 1}^n \paren {\dfrac {H_{k + 1} } {k + 1} - \dfrac 1 {\paren {k + 1} \paren {k + 1} } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 1}^n \dfrac {H_{k + 1} } {k + 1} - \sum_{k \mathop = 1}^n \dfrac 1 {\paren {k + 1}^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 2}^{n + 1} \dfrac {H_k} k - \sum_{k \mathop = 2}^{n + 1} \dfrac 1 {k^2}\) | Translation of Index Variable of Summation | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 1}^{n + 1} \dfrac {H_k} k - \dfrac {H_1} 1 - \paren{\sum_{k \mathop = 1}^{n + 1} \dfrac 1 {k^2} - \dfrac 1 {1^2} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 1}^{n + 1} \dfrac {H_k} k - 1 - \paren{\sum_{k \mathop = 1}^{n + 1} \dfrac 1 {k^2} - 1}\) | Harmonic Number $H_1$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 1}^{n + 1} \dfrac {H_k} k - \sum_{k \mathop = 1}^{n + 1} \dfrac 1 {k^2}\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 1}^{n + 1} \dfrac {H_k} k - \harm 2 {n + 1}\) | Definition of General Harmonic Numbers | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac { {H_{n + 1} }^2 + \harm 2 {n + 1} } 2 - \harm 2 {n + 1}\) | Summation to n of kth Harmonic Number over k | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac { {H_{n + 1} }^2 + \harm 2 {n + 1} - 2 \harm 2 {n + 1} } 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac { {H_{n + 1} }^2 - \harm 2 {n + 1} } 2\) |
Hence the result.
$\blacksquare$
Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.7$: Harmonic Numbers: Exercise $14$