# Sequence of General Harmonic Numbers Converges for Index Greater than 1

## Theorem

Let $\map {H^{\paren r} } n$ denote the general harmonic number:

$\ds \map {H^{\paren r} } n = \sum_{k \mathop = 1}^n \frac 1 {k^r}$

for $r \in \R_{>0}$.

Let $r > 1$.

Then as $n \to \infty$, $\map {H^{\paren r} } n$ is convergent with an upper bound of $\dfrac {2^{r - 1} } {2^{r - 1} - 1}$.

## Proof

Let $m \in \N$ be arbitrary.

Then:

 $\ds \harm r {2^{m - 1} }$ $=$ $\ds \harm r {2^{m - 1} - 1} + \dfrac 1 {\paren {2^{m - 1} }^r} + \dfrac 1 {\paren {2^{m - 1} + 1}^r} + \cdots + \dfrac 1 {\paren {2^{m - 1} + \paren {2^{m - 1} - 1} }^r}$ $\ds$ $<$ $\ds \harm r {2^{m - 1} - 1} + \dfrac 1 {\paren {2^{m - 1} }^r} + \dfrac 1 {\paren {2^{m - 1} }^r} + \cdots + \dfrac 1 {\paren {2^{m - 1} }^r}$ Ordering of Reciprocals $\ds$ $=$ $\ds \harm r {2^{m - 1} - 1} + \dfrac {2^{m - 1} } {2^{\paren {m - 1} r} }$ $\ds$ $=$ $\ds \harm r {2^{m - 1} - 1} + \paren {2^{1 - r} }^{m - 1}$ $\ds$ $<$ $\ds \harm r {2^{m - 2} - 1} + \paren {2^{1 - r} }^{m - 2} + \paren {2^{1 - r} }^{m - 1}$ $\ds$ $<$ $\ds \dots$ $\ds$ $<$ $\ds \harm r {2^0 - 1} + \paren {2^{1 - r} }^0 + \paren {2^{1 - r} }^1 + \dots + \paren {2^{1 - r} }^{m - 1}$ $\ds$ $=$ $\ds 0 + \sum_{k \mathop = 0}^{m - 1} \paren {2^{1 - r} }^k$ $\ds$ $=$ $\ds \dfrac {1 - \paren {2^{1 - r} }^m} {1 - 2^{1 - r} }$ 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 $\harm r n$ is bounded from above by $\dfrac {2^{r - 1} } {2^{r - 1} - 1}$.

By the Monotone Convergence Theorem, as $n \to \infty$, $\harm r n$ is convergent with an upper bound of $\dfrac {2^{r - 1} } {2^{r - 1} - 1}$.

$\blacksquare$