Sequence of General Harmonic Numbers Converges for Index Greater than 1
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Theorem
Let $H_n^{\paren r}$ denote the general harmonic number:
- $\ds H_n^{\paren r} = \sum_{k \mathop = 1}^n \frac 1 {k^r}$
for $r \in \R_{>0}$.
Let $r > 1$.
Then as $n \to \infty$, $H_n^{\paren r}$ is convergent with an upper bound of $\dfrac {2^{r - 1} } {2^{r - 1} - 1}$.
Proof
For any $m \in \N$:
\(\ds H_{2^m - 1}^{\paren r}\) | \(=\) | \(\ds H_{2^{m - 1} - 1}^{\paren r} + \dfrac 1 {\paren {2^{m - 1} }^r} + \dfrac 1 {\paren {2^{m - 1} + 1}^r} + \cdots + \dfrac 1 {\paren {2^{m - 1} + \paren {2^{m - 1} - 1} }^r}\) | ||||||||||||
\(\ds \) | \(<\) | \(\ds H_{2^{m - 1} - 1}^{\paren r} + \dfrac 1 {\paren {2^{m - 1} }^r} + \dfrac 1 {\paren {2^{m - 1} }^r} + \cdots + \dfrac 1 {\paren {2^{m - 1} }^r}\) | Ordering of Reciprocals | |||||||||||
\(\ds \) | \(=\) | \(\ds H_{2^{m - 1} - 1}^{\paren r} + \dfrac {2^{m - 1} } {2^{\paren {m - 1} r} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds H_{2^{m - 1} - 1}^{\paren r} + \paren {2^{1 - r} }^{m - 1}\) | ||||||||||||
\(\ds \) | \(<\) | \(\ds H_{2^{m - 2} - 1}^{\paren r} + \paren {2^{1 - r} }^{m - 2} + \paren {2^{1 - r} }^{m - 1}\) | ||||||||||||
\(\ds \) | \(<\) | \(\ds \dots\) | ||||||||||||
\(\ds \) | \(<\) | \(\ds H_{2^0 - 1}^{\paren r} + \paren {2^{1 - r} }^0 + \paren {2^{1 - r} }^1 + \dots + \paren {2^{1 - r} }^{m - 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 0 + \sum_{k \mathop = 0}^{m - 1} \paren {2^{1 - r} }^k\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {1 - \paren {2^{1 - r} }^m} {1 - 2^{1 - r} }\) | Sum of Geometric Sequence | |||||||||||
\(\ds \) | \(<\) | \(\ds \dfrac 1 {1 - 2^{1 - r} }\) | as $0 < 2^{1 - r} < 1$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {2^{r - 1} } {2^{r - 1} - 1}\) |
Since $m$ is arbitrary, every partial sum $H_n^{\paren r}$ is bounded from above by $\dfrac {2^{r - 1} } {2^{r - 1} - 1}$.
By Monotone Convergence Theorem, as $n \to \infty$, $H_n^{\paren r}$ is convergent with an upper bound of $\dfrac {2^{r - 1} } {2^{r - 1} - 1}$.
$\blacksquare$
Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.7$: Harmonic Numbers: $(4)$
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.7$: Harmonic Numbers: Exercise $3$