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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.

3
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0answers
26 views

How to evaluate $\sum_{n=1}^{\infty}\frac{H_{kn}^2-[\gamma+\ln(kn)]^2}{n}?$

From this post @Olivier Oloa gives the closed form for this sum $(1)$ $$\sum_{n=1}^{\infty}\frac{H_n^2-(\gamma+\ln n)^2}{n}=\frac{5}{3}\zeta(3)-\frac{2}{3}\gamma^3-2\gamma\gamma_1-\gamma_2\tag1$$ I ...
5
votes
2answers
138 views

Evaluating $\sum_{n=1}^\infty\frac{(H_n)^2}{n}\frac{\binom{2n}n}{4^n}$

Question: How can we evaluate $$\sum_{n=1}^\infty\frac{(H_n)^2}{n}\frac{\binom{2n}n}{4^n},$$where $H_n=\frac11+\frac12+\cdots+\frac1n$? Quick Results This series converges because $$\frac{(H_n)^2}{n}\...
0
votes
1answer
79 views

Prove that the limit of the sum is $\ln(2)$ [on hold]

I need to prove that $$\lim_{n\rightarrow\infty}\sum_{i=1}^{n}\frac{\left(-1\right)^{i+1}}{i}=\ln2$$ There is a hint I can use that says $$ \sum_{i=1}^{2n}\frac{\left(-1\right)^{i+1}}{i}=\sum_{i=...
0
votes
0answers
27 views

On the limit of $H_n^p\over n^q$ and $H_{n^p}\over n^q$

As you know, $H_n$ is the famous Harmonic number defined as follows:$$H_n=\sum_{k=1}^{n}{1\over k}$$I was wondering for which $p,q\in \Bbb N$ do the following limits exist and are equal to some ...
4
votes
2answers
133 views

Evaluate $\sum_{n=1}^{\infty} {\frac1{n} (H_{2n}-H_{n}-\ln2)}$

By accident, I find this summation when I pursue the particular value of $-\operatorname{Li_2}(\tfrac1{2})$, which equals to integral $\int_{0}^{1} {\frac{\ln(1-x)}{1+x} \mathrm{d}x}$. Notice this ...
2
votes
0answers
96 views

A sort of Wolstenholme's $p^2$-congruence.

It is well-known (Wolstenholme Theorem) that for any prime $p$ such that $p>3$ the Harmonic number $H_{p-1}$ satisfies the congruence $$ H_{p-1}:= \sum_{i=1}^{p-1}\frac{1}{i}\equiv 0 \pmod {p^2}$$ ...
2
votes
3answers
65 views

How can i prove this $\lim\limits_{n\to \infty}\dfrac{\sum_{k=1}^{n} \left(\frac1k\right)}{\sum_{k=1}^{n}{\sin \left(\frac1k\right)}}=1$?

I have accrossed the following sum in my textbook $\lim\limits_{n\to \infty}\dfrac{\sum_{k=1}^{n} \left(\frac1k\right)}{\sum_{k=1}^{n}{\sin \left(\frac1k\right)}}=1$ , I have tried to evaluate ...
2
votes
1answer
77 views

How to Prove Integral Representations of $H_z$ and $\psi(z)$

I've run across several integral representations of $H_z$ such as the following (see Harmonic Number Integral Representations). (1) $\quad H_z=\int\limits_0^1 \frac{1-t^z}{1-t} \, dt\,,\qquad\qquad\...
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votes
1answer
32 views

Elementary number th. Harmonic sum and its mod

Let $\frac{1}{2}+\frac{1}{3}+...+\frac{1}{121}=\frac{p}{q}$ where p,q are coprime integer couple. Prove $p\equiv 50 \pmod {121}.$ What im guessing about is that wolstenholme will do its job here but ...
1
vote
0answers
30 views

Leave-one-out Harmonic number

I am trying to find a number sequence $x_k\geq 0$ such that $$\text{for all integers }n\geq k\geq 1,\quad H_{n-1 - x_k(n-k)}\geq H_n-\frac{1}{k}.$$ I recall that the harmonic numbers $s\mapsto H_s$ ...
3
votes
0answers
87 views

Evaluating $\int_{0}^{1-x} \frac{1}{1-x_{n}} \cdots \int_{0}^{1-x_3} \frac{1}{1-x_2} \int_{0}^{1-x_2} \frac{1}{1-x_1} dx_1 dx_2 \cdots dx_n$

I am trying to evaluate the following integral $$\int_{0}^{1-x} \frac{1}{1-x_{n}} \cdots \int_{0}^{1-x_3} \frac{1}{1-x_2} \int_{0}^{1-x_2} \frac{1}{1-x_1} dx_1 dx_2 \cdots dx_{n}, \hspace{0.5cm} 0<...
1
vote
2answers
90 views

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 ...
1
vote
1answer
42 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 ...
0
votes
0answers
28 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 ...
0
votes
0answers
27 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
votes
1answer
109 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) \...
-3
votes
1answer
68 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
votes
4answers
806 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}(\...
0
votes
3answers
98 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)}\...
2
votes
2answers
34 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 + {...
-4
votes
1answer
105 views

Harmonic number and the golden ratio

I just don't understand how $(1)$ can have this simple closed form. $$\sum_{k=1}^{\infty}\frac{{2k \choose k}}{(-16)^k}[H_k-H_{k+1}]=\phi^{-6}\tag1$$ Where $\phi=\frac{1+\sqrt{5}}{2}$, is the ...
2
votes
0answers
49 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}^...
4
votes
1answer
55 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
votes
1answer
53 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(...
8
votes
2answers
793 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, ...
1
vote
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 ...
2
votes
1answer
59 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 ...
0
votes
2answers
56 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 ...
8
votes
1answer
131 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)(...
3
votes
2answers
95 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 ...
6
votes
2answers
346 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. ...
13
votes
4answers
181 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}...
0
votes
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
votes
1answer
238 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 ...
0
votes
0answers
85 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 ...
0
votes
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}+\...
1
vote
1answer
64 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.
0
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3answers
187 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 ...
2
votes
3answers
90 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} + \...
0
votes
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\...
6
votes
1answer
108 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}...
0
votes
0answers
55 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 ...
1
vote
1answer
54 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
votes
2answers
42 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 ...
1
vote
2answers
61 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
votes
3answers
142 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 ...
1
vote
1answer
30 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 ...
4
votes
1answer
78 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{\...
0
votes
1answer
99 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 ...
1
vote
0answers
43 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}{...