Sequence of General Harmonic Numbers Converges for Index Greater than 1
Theorem
Let $H_n^{\left({r}\right)}$ denote the general harmonic number:
- $\displaystyle H_n^{\left({r}\right)} = \sum_{k \mathop = 1}^n \frac 1 {k^r}$
for $r \in \R_{>0}$.
Let $r > 1$.
Then as $n \to \infty$, $H_n^{\left({r}\right)}$ is convergent with an upper bound of $\dfrac {2^{r - 1} } {2^{r - 1} - 1}$.
Proof
| \(\displaystyle H_{2^m}^{\left({r}\right)}\) | \(=\) | \(\displaystyle H_{2^{m - 1} }^{\left({r}\right)} + \dfrac 1 {\left({2^{m - 1} + 1}\right)^r} + \dfrac 1 {\left({2^{m - 1} + 2}\right)^r} + \cdots + \dfrac 1 {\left({2^{m - 1} + 2^m}\right)^r}\) | |||||||||||
| \(\displaystyle \) | \(\le\) | \(\displaystyle H_{2^{m - 1} }^{\left({r}\right)} + \dfrac 1 {\left({2^{m - 1} }\right)^r} + \dfrac 1 {\left({2^{m - 1} }\right)^r} + \cdots + \dfrac 1 {\left({2^{m - 1} }\right)^r}\) | |||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle H_{2^{m - 1} }^{\left({r}\right)} + \dfrac {2^m} {2^{\left({m - 1}\right) r} }\) | |||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle H_{2^{m - 1} }^{\left({r}\right)} + \dfrac 1 {2^{m \left({r - 1}\right)} }\) | |||||||||||
| \(\displaystyle \implies \ \ \) | \(\displaystyle H_{2^{m + 1} }^{\left({r}\right)}\) | \(\le\) | \(\displaystyle \sum_{0 \mathop \le k \mathop < m} \dfrac {2^k} {2^{k r} }\) | ||||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \dfrac {2^{r - 1} } {2^{r - 1} - 1}\) | Sum of Geometric Progression |
$\blacksquare$
I wish I wasn't so tired.
You can help ProofWiki by reviewing it, and either correcting it or adding some explanatory material as to why you believe it is actually correct after all.
To discuss this point in more detail, feel free to use the talk page.
If you are able to resolve this, then when you have done so you can remove this instance of {{Mistake}} from the code.
If you would welcome a second opinion as to whether your explanation is correct, add a call to {{Proofread}} the page (see the proofread template for usage).
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 $3$
