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.

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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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1answer
50 views

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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33 views

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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28 views

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 ...
8
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1answer
124 views

Two Euler sums each containing the reciprocal of the central binomial coefficient

Is it possible to find closed-form expressions for the following two Euler sums containing the reciprocal of the central binomial coefficient? $$1. \sum_{n = 0}^\infty \frac{(-1)^n H_n}{(2n + 1) \...
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1answer
69 views

Is there a formula for a summation divided by a product of its terms?

$$\frac{\sum_{i=1}^{n}x_{i}}{\prod_{i=1}^{n}x_{i}}= \frac{1}{x_{2}x_{3}x_{4}...}+\frac{1}{x_{1}x_{3}x_{4}}+\frac{1}{x_{1}x_{2}x_{4}}...$$ There is a very clear pattern that each consecutive result ...
5
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4answers
817 views

determining if sequence has upper bound

I am somewhat stuck in my calculations when determining if sequence has an upper bound. The sequence $$x_n = \frac{1}{n+1}+\frac{1}{n+2}+..+\frac{1}{2n-1}+\frac{1}{2n}$$ Is equal to $$\frac{1}{n}(\...
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3answers
149 views

New formula for complex harmonic progression

If $a$ is integer and $\textbf{i} b$ is not integer then: $\sum_{k=1}^{n}\frac{1}{a i k+b}=-\frac{1}{2b}+\frac{1}{2(a i n+b)}+\frac{2\pi}{e^{2\pi b}-1}\int_{0}^{1}e^{\pi(a i n+2b)u}\sin{(\pi a n u)}\...
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2answers
37 views

How many terms are required for harmonic series of degrees to “cover” a full $360^\text{o}$ circle?

This question accidentally came to my mind when reading about harmonic series. I've never been able to find an answer on the Internet. Consider $H_n$ which is the $n$-th harmonic number: $$ H_n = 1 + {...
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0answers
53 views

Sum with harmonic number of squared argument as e.g. $\sum_{k=1}^\infty \frac{H(k^2)}{k^2}$

I wonder if closed expression can be found for sums of harmonic numbers with a squared argument. Examples are $$s_{1}=\sum_{k=1}^\infty \frac{ H(k^2)}{k^2} \simeq 3.28709\tag{1}$$ $$s_{2}=\sum_{k=1}^...
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1answer
56 views

What is the asymptotic form for the sum of the reciprocals of the first $n$ primes?

It is well known that the sum of the reciprocals of the primes $p$ less than or equal to a maximum value $n$ is asymptotic to $\ln \ln n$: $$\sum_{p \leq n} \frac{1}{p} \sim \ln \ln n.$$ (The next ...
2
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1answer
54 views

Sums of Reciprocals of Polynomials and Harmonic Numbers

This is question based on a pattern I have noticed while using mathematica. Let $P(x)$ be a polynomial with real, simple, negative roots $r_n$ ($n:1,2,...,k$) and define $$Q_n=\lim_{x\to r_n}\frac{P(...
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2answers
798 views

Is the sequence of sums of inverse of natural numbers bounded? [duplicate]

I'm reading through Spivak Ch.22 (Infinite Sequences) right now. He mentioned in the written portion that it's often not a trivial matter to determine the boundedness of sequences. With that in mind, ...
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1answer
34 views

Representation for harmonic series $H_n$, for $n<-1$

According to Wolfram Alpha, harmonic series $H_x$ has the following representation: $$H_x=\int_{0}^{1}\frac{-1+t^x}{-1+t}dt=\int_{0}^{\infty}\frac{1-e^{-xt}}{-1+e^t}dt,~Re(x)>-1$$ The ...
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1answer
65 views

Logarithmic integrals and Euler sums

At various places e.g. Calculate $\int_0^1\frac{\log^2(1+x)\log(x)\log(1-x)}{1-x}dx$ and How to prove $\int_0^1x\ln^2(1+x)\ln(\frac{x^2}{1+x})\frac{dx}{1+x^2}$ logarithmic integrals are connected ...
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2answers
57 views

Is there a way to sum (part of) the harmonic series to a given total?

For example, $1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6$ can be solved by traditional arithmetic of $1 + 360/720 + 240/720...$ etc. to get a total of $2.45$, but if I wanted the total to say, $1/12367$, the ...
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1answer
149 views

Generalizing $\sum_{n=1}^{\infty}\frac{H_n{2n \choose n}}{2^{2n}(2n-1)}=2$

I was looking at this paper on section [17], $$\sum_{n=1}^{\infty}\frac{H_n{2n \choose n}}{2^{2n}(2n-1)}=2\tag1$$ Let generalize $(1)$ $$\sum_{n=1}^{\infty}\frac{H_n{2n \choose n}}{2^{2n}(2n-1)(...
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2answers
96 views

Closed expression for alternating harmonically wrapped harmonic series $\sum_{n=1}^\infty (-1)^{n+1} H_{\frac{1}{n}}$

It is well known that the alternating harmonic sum $\sum_{n=1}^\infty (-1)^{n+1} \frac{1}{n}$ converges to $\log(2)$. Now let us wrap $\frac{1}{n}$ with the harmonic number $H_k$ (continued ...
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2answers
383 views

Analytic continuation of harmonic series

Is there an accepted analytic continuation of $\sum_{n=1}^m \frac{1}{n}$? Even a continuation to positive reals would be of interested, though negative and complex arguments would also be interesting. ...
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4answers
191 views

Showing $\sum_{k=1}^{nm} \frac{1}{k} \approx \sum_{k=1}^{n} \frac{1}{k} + \sum_{k=1}^{m} \frac{1}{k}$

Since $\log(nm) = \log(n) + \log(m)$, and $\sum_{k=1}^n \frac{1}{k} \approx \log n$ for large $n$, we would expect that $$\sum_{k=1}^{nm} \frac{1}{k} \approx \sum_{k=1}^{n} \frac{1}{k} + \sum_{k=1}^{m}...
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1answer
51 views

Isolating $k$ in $H_k=\frac{c}{k+1}$

I am trying to find an equilibrium point of two algorithms, parametrized by $k$. The performance of the two algorithms: $\frac{c}{k+1}$ (where $c$ is some given positive constant) $H_k$ (the $k$-th ...
14
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1answer
242 views

Evaluating $1-\frac12-\frac13+\frac14+\frac15+\frac16-\cdots$

What is the value of $$S=1-\frac12-\frac13+\frac14+\frac15+\frac16-\cdots$$ where the sign alternates over the triangular numbers? To start, it is easy to prove convergence. The sum of each set of ...
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0answers
86 views

Evaluating the continued fraction

How to evaluate $\frac{1}{a_{1}+\frac{1}{a_{2}+\frac{1}{a_{3}+\frac{1}{a_{4}+\frac{1}{\ddots}}}}}$, where $a_{j}=\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\dots+\frac{1}{j}$? That is, How to evaluate ...
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0answers
36 views

This semi-harmonic-series converges [duplicate]

We know that the series $\sum_{k=1}^{\infty}\frac{1}{k}$ diverges. The series $\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\dots+\frac{1}{8}+\frac{1}{10}+\frac{1}{11}+\dots+\frac{1}{18}+\frac{1}{20}+\...
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1answer
68 views

Evaluation of Euler - type sum $\sum\limits_{n=1}^{\infty}\frac{\left(H_{2n}-\frac{1}{2}H_{n}\right)^{2}}{n^{2}}$ [closed]

How can one evaluate the sum $\displaystyle\sum_{n=1}^{\infty}\frac{\left(H_{2n}-\frac{1}{2}H_{n}\right)^{2}}{n^{2}}$? Here $H_{n}$ denotes the $n$-th harmonic number.
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3answers
203 views

Closed form for $1+\frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n}$? [duplicate]

Background When I attempt questions in real analysis, frequently I encounter the following expression like harmonic series: $$H_n=1+\frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n}$$ To my surprise ...
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3answers
91 views

Is $1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{x} < \log_2 x$

If $x \ge 5$, is $1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{x} < \log_2 x$ I believe the answer is yes. Here is my thinking: (1) $\log_2{5} > 2.32 > 2.284 > 1 + \frac{1}{2} + \...
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0answers
7 views

Find the least integer $k$ such that $\left[H_k\right]=n$ for a given positive integer $n$

At the moment I am curious about harmonic numbers at their properties and came across the following question. We are given $n\in\mathbb{N}$. We are to find the least integer $k$ such that $\left[H_k\...
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1answer
123 views

Prove $\sum_{k=1}^{\infty}\frac{(-1)^k}{(2k+1)^2}(2H_{2k}+H_k)=\frac{\pi^3}{32}-2G\ln2$

How to prove $$\sum_{k=1}^{\infty}\frac{(-1)^k}{(2k+1)^2}(2H_{2k}+H_k)\stackrel ?=\frac{\pi^3}{32}-2G\ln2,$$ where $G$ is the Catalan's constant. Attempt For the first sum, $$\sum_{k=1}^{\infty}...
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0answers
70 views

Partial sum of general harmonic series

I have a trouble to calculate or find accurate approximation for the following finite series: $$\sum_{i=1}^{n} \frac {1} {(1+s+id)^i}.$$ How above mentioned partial sum can be calculated or ...
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1answer
55 views

Combinatorics: Number of permutations from $1$ to $60$, with cyclic permutation of length $l > 30$

In a game, $60$ kids have the chance to win presents. The organiser gives every kid a number between $1$ and $60$, so that all kids have different numbers. In a room, there is a cupboard with $60$ ...
3
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2answers
45 views

holomorphic functions, roots of unity and harmonic numbers

If $f$ is a non-constant holomorphic function such that, for all $z \in \mathbb{C}$, exists a $c \in \mathbb{C}$ where $f(cz) = f(z),$ then $c$ must be a $n$-th root of unity, or there exists some ...
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2answers
78 views

Find a reordering of the alternating harmonic sequence such that the limit of the partial sums is $\infty$.

The alternating harmonic series is defined as normal ($a_k = \frac{(-1)^{k+1}}{k}$). Not sure how I could do this, as I would have to include all the positive and negative parts, which has a ...
3
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4answers
176 views

Can $\sum_{n=1}^\infty\frac{H_{2n}}{(2n)^3}$, with $H_n$ the $n$-th harmonic number, be written in terms of $\zeta$ values?

The Euler sums are given by $$S_{p,q} = \sum_{n = 1}^{\infty} \frac{H_{n}^{(p)}}{n^q},$$ where $$H_{n}^{(p)} = \sum_{j = 1}^{n} \frac{1}{j^p}.$$ According to Wolfram, Eq. (19), the following special ...
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1answer
40 views

Concrete Mathematics: Quicksort analysis

I'm trying to work my way through Concrete Mathematics (2nd Edition). I'm struggling to understand the transition for the analysis of the quick sort algorithm from the recurrence to the harmonic ...
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1answer
81 views

Express $\sum_{n=1}^{\infty} \dfrac1{\prod_{k=1}^m (n+a_k)} $ as a finite sum.

This is a generalization (and answer to) Writting $S = \sum_{k=0}^{\infty} \frac{1}{(r_1+k+1)(r_2+k+1)(r_3+k+1)}$ as a rational function of $r_1,r_2$ and $r_3$. Let $S =\sum_{n=1}^{\infty} \dfrac1{\...
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1answer
111 views

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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0answers
46 views

What is a common way how to optimize such kind of equations? [closed]

$$ \frac{20}{1000 }+\frac{ 19}{999 }+\frac{ 18}{998 }+\frac{ 17}{997 }+\frac{ 16}{996 }+\frac{ 15}{995 }+\frac{ 14}{994 }+\frac{ 13}{993 }+\frac{ 12}{992 }+\frac{ 11}{991 }+\frac{ 10}{990 }+\frac{ 9}{...
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1answer
56 views

Number of primes vs harmonic series

Prove that for $p_k < n < p_{k+1}$, where $p_k$ is k-th prime, $1+ 1/2 + ... 1/n < k + 1$ I am trying to use the estimate, $\pi(n) > \ln(n) - 1 $ , but cannot get to the required ...
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0answers
54 views

A very strict inequality for $\ln n$ and $H_{n-1}$, with $n>2$, searching for proof

I have been fooling around with logarithms and experimentally found a very strict inequality from above (which I haven't been able to improve further): $$\ln n < H_{n-1}+\frac{1}{8(n-1)}+\frac{3}...
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0answers
43 views

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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0answers
66 views

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 ...
9
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5answers
243 views

(Combinatorial?) Proof of the identity $\sum_{k=1}^n \frac {(-1)^k}{k\,(k+1)}\binom nk = \frac 12 + \frac 13 + \dots + \frac 1{n+1}$?

Recently I've come across an interesting identity: $$ \frac 1{1\cdot 2}\binom n1 - \frac 1{2\cdot 3}\binom n2 + \frac 1{3\cdot 4}\binom n3 - \dots + \frac {(-1)^n}{n\cdot (n+1)}\binom nn = \frac 12 + \...
4
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0answers
90 views

Closed form of finite Euler sum $\sum_{k=1}^n \frac{ H_{k}}{(2k+1)}$

Recently (Finite sums of products of harmonic numbers like $\sum_{k=1}^n H_{k} H_{2k}$) I came across this finite Euler sum $$p_{2}(n) = \sum_{k=1}^n \frac{H_{k}}{2k+1}\tag{1}$$ and I wonder if it ...
1
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1answer
84 views

Finite sums of products of harmonic numbers like $\sum_{k=1}^n H_{k} H_{2k}$

The problem In Sum of powers of Harmonic Numbers finite sums of powers of the same harmonic number have been studied: $$s(q,n) = \sum_{k=1}^n H_{k}^q\tag{1}$$ Here we turn to the related questions ...
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1answer
85 views

Is there an arithmetic mean limit on the symmetrical items of the harmonic series?

Is it possible to determine what is the arithmetic mean of the harmonic series where $n$ and $-n$ are added and divided by two in this manner: $$f(n) = \frac{ (\frac{3}{4})^n \times 2^{⌈-n \times ...
0
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2answers
32 views

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 ...
5
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2answers
131 views

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}+...
0
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0answers
31 views

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 ...
2
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1answer
60 views

Card shuffling and harmonic numbers

Below is a simple proof of one connection between card shuffling and harmonic numbers. I'm interested in references for this if it's already known, as well as alternative methods of proof. (Can the ...