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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.

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22 views

Double harmonic series $\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\frac{H_{n+m}^{(p)}}{(n+1)^{q}(m+1)^{r}}$

Do these sums exist in the literature and have been investigated before? The same question for the odd variant, that is $$ \sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\frac{O_{n+m}^{(p)}}{(2n+1)^{q}(2m+1)^{...
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32 views

A “binomial” generalization of harmonic numbers

For positive integers $s$ and $n$ (let's limit the generality), define $$H_s(n)=\sum_{k=1}^{n}\frac{1}{k^s},\qquad G_s(n)=\sum_{k=1}^{n}\binom{n}{k}\frac{(-1)^{k-1}}{k^s}.$$ The former is well-known; ...
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40 views

Expressions approximating Generalized Harmonic Number (truncated polynomials with shrinking error term preferred)

Specifically, $$H_m^{(2n)} \approx\ ?$$ and $$H_m^{(4n)} \approx\ ?$$ where $(m, n)$ $\in \mathbb N_{>1}$ I would not like to use special functions like the (Riemann zeta function) unless they ...
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2answers
57 views

Show that : $\sum_{k=1}^{\infty}\frac{i^{k(5k+1)}}{k(k+1)}=1-\frac{π}{2}$

Show that $S=\displaystyle\sum_{k=1}^{\infty}\frac{i^{k(5k+1)}}{k(k+1)}=1-\frac{π}{2}$ My try : $S=\displaystyle\sum_{k=1}^{\infty}\frac{e^{iπk(5k+1)/2}}{k(k+1)}$ $=\displaystyle\sum_{k=1}^{\...
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2answers
75 views

Compute : $\sum_{m≥1}\frac{(-1)^{m}}{(m+1)(2m+1)^{2}}$

How can Compute in closed form this double summation : $\displaystyle\sum_{m≥1}\frac{(-1)^{m}}{(m+1)(2m+1)^{2}}$ I need to evaluate this sum using digamma function Actually I don't have any ideas ...
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1answer
60 views

Compute $\sum_{k≥1,n≥1}\frac{(-1)^{n+k}}{k(k+1+n)^2}$

How can Compute in closed form this double summation : $$\sum_{k≥1,n≥1}\frac{(-1)^{n+k}}{k(k+1+n)^2}$$ I think here can use harmonic series Actually I don't have any ideas to approach it
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127 views

On binomial sums $\sum_{n=1}^\infty \frac{1}{n^k\,\binom {2n}n}$ and log sine integrals

Seven years ago, I asked about closed-forms for the binomial sum $$\sum_{n=1}^\infty \frac{1}{n^k\,\binom {2n}n}$$ Some alternative results have been made. Up to a certain $k$, it seems it can be ...
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3answers
117 views

Compute in closed form $\int_0^1\frac{\arctan{ax}}{\sqrt{1-x^{2}}}dx$

I am trying to find closed form for this integral: $$I(a)=\int_0^1\frac{\arctan{ax}}{\sqrt{1-x^{2}}}dx$$ Where $a>0$. My try: Let: $$I(a)=\int_0^1\frac{\arctan{ax}}{\sqrt{1-x^{2}}}dx$$ Then: $$\...
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2answers
22 views

Is there a further simplification for alternating harmonics with order?

I know that the harmonic number $H_a ^{(b)}$ is $$\sum_{n=1}^a \frac{1}{n^b}$$ I was wondering if, for the generalized alternating harmonic number $\bar H_a^{(b)}$, there was a closed formula. For ...
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2answers
28 views

Asymptotic analysis of harmonic series using Calculus

The problem is to proof that Harmonic series $\sum_{i=1}^n \frac{1}{i} = O(ln \space n)$ So, I know that $ln \space n = \int_{1}^n \frac{1}{x} dx$ so, I need to prove that $H(n) = 1+\frac{1}{2}+...+...
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0answers
49 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 ...
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2answers
157 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}\...
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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 ...
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2answers
138 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 ...
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0answers
97 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 $$ H_{p-1}:= \sum_{i=1}^{p-1}\frac{1}{i}\equiv 0 \pmod {p^2}$$ ...
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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
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1answer
80 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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1answer
33 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 ...
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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
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0answers
88 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<...
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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 ...
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1answer
48 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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30 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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0answers
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
112 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
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 ...
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4answers
812 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
142 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
35 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
51 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 ...
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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
796 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
64 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
142 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 ...
6
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2answers
369 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
184 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
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
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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}+\...
1
vote
1answer
67 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
195 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
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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\...
6
votes
1answer
118 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
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0answers
66 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 ...