Questions tagged [harmonic-numbers]

For questions regarding harmonic numbers, which are partial sums of the harmonic series. The $N$-th harmonic number is the sum of reciprocals of the first $N$ natural numbers.

588 questions
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Do these polynomials with harmonic number-related coefficients lie in some particular known class?

I've generated a set of univariate polynomials ($b=1,2,\ldots$) in $v$ of degree $b-1$. The constant term and the coefficient of $v^{b-1}$ is simply $H_b$, the $b$-th harmonic number. The ...
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Estimate for multiple harmonic sum

I am interested in estimating the following family of sums: $$S_k(n) \equiv \sum_{\substack{n_1, \ldots, n_k \geq 1\\n_1 + \ldots + n_k = n}}\frac{1}{n_1\ldots n_k}$$ where $k \geq 1, n \geq 1$. A ...
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Harmonic series (double)

I am wondering about the $\Theta$ class (i.e. asymptotic complexity) of the following: $$\sum^n_{i=1} \sum_{j=1}^n \frac{1}{ij}$$ Since this is basically the harmonic series, applied twice, it ...
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Harmonic numbers

Is there a formula for calculating such a sequence of numbers: 1/1 + 1/2 + 1/3 + 1/4 + 1/5 + ... 1/x? I know that the sum of the Harmonic series is equal to infinity, but is there a formula for ...
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Sum to n terms of the harmonic series [duplicate]

We know that $\sum_{k=1}^{\infty}\frac{1}{k}=\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\cdots$ diverges. But for any natural number $n$, $\sum_{k=1}^{n}\frac{1}{k}$ is finite. The question is; how to ...
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Question on the difference between harmonic series.

I found the following identity: For $r>s>0$ and integers. \begin{align} \sum_{m=1}^{\infty} \frac{1}{m+r} \frac{1}{m+s} = \frac{H_r-H_s}{r-s} = \frac{1}{r-s} \left( \frac{1}{s+1} + \frac{1}{s+2}...
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Why can't the divergence of the harmonic series be resolved?

Calculus 1 students are taught that some infinite series converge while others do not. However, there are other tricks one can do whereby some of those "diverging" series do in fact converge but in ...
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What is the solution of normalized harmonic series based on $4/3$ between one and two?

In music theory notes generated by the consequencing interval of $4/3$ generates harmonic series. Series can be normalized by multiplicating the fraction with a $2$ in power $n$. What is a formula ...
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Constant term in Stirling type formula for $\sum^N_{n=1} H_n \cdot \ln(n)$

Is there a similar formula like the Stirling one on the sum over $\ln(n)$ (take logarithms on its factorial representation),  \sum_{n=1}^N \ln(n) = N\cdot \ln(N)−N+\frac{\ln(N)}{2}+\frac{\ln(2π)}{2}+...
Infinite sum of Harmonic Numbers of order r: $\sum_{k=1}^{\infty}H_r(k)k^{-s}$
The Wolfram site lists a formula for $\sum_{k=1}^{\infty}\frac{H_r(k)}{k^r}$, where $H_r(k)$ is the Harmonic Number of order $r$, that is, $H_r(k)=\sum_{i=1}^{k}i^{-r}$. This formula is below to aid ...