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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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155 views

How to prove this question by Ramanujan?

click here for photo $$1+2\sum_{k=1}^\infty \frac{1}{(4k)^3-(4k)}= \frac{3}{2}\ln(2)\,.$$ well i have attatched a photo which has been asked to prove without using calculus,but how to solve this ...
3
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2answers
93 views

For any $n \in \mathbb{N}$, show that: $\frac{1}{n+1} + \frac{1}{n+2} + \ldots + \frac{1}{2n} < \frac{5}{6}$.

For any $n \in \mathbb{N}$, show that: $$\frac{1}{n+1} + \frac{1}{n+2} + \ldots + \frac{1}{2n} < \frac{5}{6}$$ I wrote the sum as $H_{2n} - H_{n}$, where $H_{k} = \frac{1}{1} + \frac{1}{2} + \...
2
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0answers
89 views

Asymptotic behaviour of Schatten $p$-norm

Consider the following matrix $$\Gamma(4) = \begin{pmatrix} 0 & 1 & 0 & 0\\ -1 & 0 & \sqrt{2} & 0\\ 0 & -\sqrt{2} & 0 & \sqrt{3}\\ 0 & 0 & -\sqrt{3} &0 ...
2
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4answers
102 views

Prove that $1+\frac{1}{2}+\frac{1}{3}+…+\frac{1}{2^n-1}<n$ for $n\geq{2}$

I tried using mathematical induction to prove this, but the problem I faced was that there are a lot of numbers between $\frac{1}{2^k-1}$ and $\frac{1}{2^{k+1}-1}$. Is it possible to prove this with ...
2
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0answers
69 views

Mellin Transform

If $$ H(x)=\sum_{n \leq x} \frac{1}{n} $$ what is its Mellin transform? I was able to find the Mellin transform of $\log(x+1)$ and of $\frac{1}{x+1}$, but I'm quite a bit inexperienced so I haven't ...
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7answers
143 views

Prove that $\lim\limits_{n\to\infty}\frac{1}{n}\left(1+\frac{1}{2}+\cdots+\frac{1}{n}\right)=0$ [duplicate]

I'm preparing for exam and I have to solve this problem. Please, how do I prove that $\lim\limits_{n\to\infty}\frac{1}{n}\left(1+\frac{1}{2}+\cdots+\frac{1}{n}\right)=0.$ Any Theorems to guide as ...
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3answers
174 views

random harmonic series

Let $(X_{n})_{n \in \mathbb{N}}$ be independent with Rademacher distribution: \begin{equation} \mathbb{P}(X_{n} = -1) = \frac{1}{2} = \mathbb{P}(X_{n} = 1). \end{equation} I have to investigate \...
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2answers
66 views

Approximation to a partial sum

Just like $$\sum_{k=1}^{n}\frac1k$$ can be approximated by log(n), is there a similar approximation for the sum $$\sum_{k=1}^{n}\frac1{k^2}$$
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2answers
53 views

Is there a reason why this function does not exist/can't be found?

I'm looking at a function $f\colon \mathbb N \rightarrow \mathbb R$, defined such that $(\Delta f)(x) = 1/x$. However, I know such a function does not exist or has not been found yet. I'm interested ...
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2answers
73 views

Why is $\left(\frac{1}{y} + \frac{1}{z}\right)\left(\frac{1}{z} + \frac{1}{x}\right)\left(\frac{1}{x} + \frac{1}{y}\right) \geq \frac{8}{xyz}$?

The solution of a question in my book uses the property that $$\left(\frac{1}{y} + \frac{1}{z}\right)\left(\frac{1}{z} + \frac{1}{x}\right)\left(\frac{1}{x} + \frac{1}{y}\right) \geq \frac{8}{xyz}$$ ...
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1answer
67 views

Sum series product

in a probability computation problem on research, I am facing a conditional probability, which can be modeled with the following simplified formulation: $\sum_{x=0}^{\infty} \frac{1}{(x-a)(x-b)}$, $a&...
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2answers
164 views

General formula for $\frac{1}{1} + \frac{1}{2}+\frac{1}{3} … +\frac{1}{n} $ [closed]

We know that this series does not converge and tends to infinity but is there a general and exact formula for sum to n terms of this series
4
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2answers
111 views

Analytic properties of Euler sums

Introduction As far as I know this topic has not been discussed before. I have found interesting results which I wish to share with you in the standard MSE manner by asking questions. Consider the ...
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2answers
163 views

Relation between harmonic series $H(m)$ and polygamma function?

I have the following formula: $$h(x)=\sum_{R=1}^m \frac{1}{R+1}-1$$ I have re-expressed this (correctly, I hope!) in terms of the harmonic number $H(m)=\sum_{R=1}^m \frac{1}{R}$: $$h(x)=H(m)+\frac{...
2
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1answer
63 views

How do we rearrange the terms of the harmonic series so they add up to 0

The harmonic series converges conditionally; therefore, the series can be rearranged in any way to get different sums, but how do we rearrange in such a way that it equates to 0. Is there a trick/ ...
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2answers
337 views

If $a$, $b$, $c$ are three positive numbers in harmonic sequence, show that $a^2+c^2>2b^2$.

If $a$, $b$, $c$ are three positive numbers in harmonic sequence, show that $a^2+c^2>2b^2$. My attempt : $(a-c)^2>0$ $a^2+c^2>2ac$ This is true for any two real positive values of $a$ and $...
1
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1answer
59 views

Problem involving harmonic number

The problem is as follows and I have trouble solving part $(b)$: I attempted the problem in the following way: Let $d(i)$ be the total length covered before the end of $i$th second. Then $d(i)=d(i-1)...
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5answers
238 views

How to show $\sum_{n=1}^{\infty}\frac{H_{n+1}}{n(n+1)}=2$?

I am interested in the proof of $$\sum_{n=1}^{\infty}\frac{H_{n+1}}{n(n+1)}=2, \quad H_{n}=1+\frac{1}{2}+\cdots+\frac{1}{n}$$ This result can be verified by Mathematica or by WolframAlpha This ...
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1answer
125 views

Elementary proof of $\sum_{k=1}^{n}\frac{1}{2k+1}$ with all odd denominators is never an integer?

For all natural number $n$, $$ \frac{1}{3} + \frac{1}{5}+...+\frac{1}{2n+1}$$ is never an integer. I know that $\sum_{k=1}^{n}\frac{1}{k}$ is never an integer, but I'm not sure how to restrict that ...
3
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2answers
83 views

Asymptotics of product of harmonic numbers

Consider the product $$p_n = \prod_{k=1}^n H_k$$ of $n$ successive harmonic numbers $H_k=\sum_{i=1}^k 1/i$. The sequence of the $p_n$ is listed in OEIS as ...
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0answers
81 views

Series involving product of Harmonic Numbers

Are there any identities involving this product of Harmonic Numbers? $$u_{n,m}=\prod_{k=1}^{n-1} H_k \prod_{k=1}^m H_{k+n}$$ My motivation is an attempt to study a series of the form $$\sum_{n=1}^\...
3
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4answers
675 views

Derivative of binomial coefficients

I obtained the following formula in Mathematica: $$\frac{d}{dn}\ln\binom{n}{k} = H_{n} - H_{n-k}$$ where $H_n$ are the harmonic numbers ($H_n = \sum_{i=1}^n 1/i$). But I have no idea how to prove it....
2
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3answers
93 views

Binomial Harmonic Numbers

Prove this equation for $0 \leq m \leq n$: $$ \frac{1}{\binom{n}{m}}\sum_{k=1}^m \binom{n-k}{n-m} \frac{1}{k} = H_n - H_{n-m} $$ where $H_k$ denotes the k-th harmonic number $\left(~H_k := \sum_{n=1}^...
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3answers
461 views

Calculating the summation$\sum_{n=1}^{\infty}\frac{H_{n+1}}{n(n+1)}$

I need to find explicitly the following summation $$\sum_{n=1}^{\infty}\frac{H_{n+1}}{n(n+1)}, \quad H_{n}=1+\frac{1}{2}+\cdots+\frac{1}{n}$$ From Mathematica, I checked that the answer is $2$. The ...
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0answers
39 views

first derivative of exponential generating function of harmonic numbers

I'm trying to prove that the the first derivative of the exponential generating function of the Harmonic numbers, $H_n$, is the exponential generating function of $H_{n+1}$, but I my solution seems ...
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1answer
33 views

Show that for $n\gt 2$, $\frac{\sigma_1(n)}{n}\lt H_n$

Is there any positive integer $n$, besides $n=2$ such that $$\frac{\sigma_1(n)}{n}=H_n$$ They are clearly asymptotic from their graphs so can we show that for $n\gt 2$, $$\frac{\sigma_1(n)}{n}\lt H_n$...
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0answers
64 views

Justify an approximation of $-\sum_{n=2}^\infty H_n\left(\frac{1}{\zeta(n)}-1\right)$, where $H_n$ denotes the $n$th harmonic number

For integers $n\geq 1$, let $\mu(n)$ the Möbius function, see this MathWorld, and let $H_n$ the $n$th harmonic number as $H_n=1+1/2+\ldots+1/n$. Also we denote the Riemann zeta function with $\zeta(s)$...
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3answers
51 views

Limit of the difference between two harmonic numbers

I am trying to evaluate the following limit: $$ \lim_{n \rightarrow \infty} H_{kn} - H_n = \lim_{n \rightarrow \infty} \sum_{i = 1}^{(k - 1)n} \frac{1}{n + 1} $$ I know that the answer is $\ln k$, ...
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0answers
38 views

Simple formula for $H_n = m + \alpha $?

Let $H_i$ be the $i$ th harmonic number. For a given positive integer $m$ we want to find the smallest possible positive integer value $n$ such that $H_n = m + \alpha $, where $\alpha > 0$. Let us ...
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0answers
44 views

On integer sequences of the form $\sum_{n=1}^N (a(n))^2H_n^2,$ where $H_n$ is the $n$th harmonic number: refute my conjecture and add yourself example

After I've read a problem posed by Édouard Lucas from the first paragraph of the biography from this Wikipedia's article I am trying to get different variations of this problem using number theoretic ...
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0answers
37 views

A non obvious example of a sequence $a(k)\cdot H_{b(k)}$ whose general term is integer many times, where $H_n$ denotes the $n$th harmonic number

For integers $n\geq 1$, $H_n$ denotes the $n$th harmonic number. I am looking examples of arithmetic function $a(k)$ and $b(k)$, whose terms are integers $\geq 1$ and such that the terms of this ...
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1answer
112 views

Infinite series with harmonic numbers related to elliptic integrals

It is known that the following functions are elliptic integrals $$ \, _2F_1\left({a,1-a\atop 1};x\right),\quad a=\tfrac12,\tfrac13,\tfrac14,\tfrac16,\tag{1} $$ $$ \, _2F_1\left({\tfrac{1}{3},\tfrac{2}{...
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1answer
92 views

Hypergeometric series with harmonic factor

I am following up on a post I made a couple months ago as I am revisiting this problem. I desire a way to approximate the sum $$\sum_{n\geq 0}\frac{\binom{2n}{n}^2 z^n}{16^n}H_n$$ for a specified ...
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2answers
2k views

A Gift Problem for the Year 2018 [duplicate]

We had this problem in exam class yesterday on Combinatoric and it was supposed to be the new year gift from our teacher. The exercise was entitled A Gift Problem for the Year 2018 Problem: ...
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0answers
35 views

Is there any formulas, which gives partial sum of $H_{n}$ as rational fraction? [closed]

If $H_{n}$ - partial sum of harmonic number, then $$H_{2}=1+\frac{1}{2}=\frac{3}{2}, H_{3}=\frac{3}{2}+\frac{1}{3}=\frac{11}{6}$$ Is there any formulas, which gives this partial sum as rational ...
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1answer
68 views

Sum, series and Harmonic numbers [duplicate]

I found the solution of series on Wolfram Alpha http://www.wolframalpha.com/input/?i=sum+1%2F(2k%2B1)%2F(2k%2B2)+from+1+to+n%2F2 for $\alpha = 1$ $\sum\limits_{k=1}^{n/2} \left(\frac{1}{\left(2k-1+2^...
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2answers
76 views

Symmetry relation for finite sums of generalized harmonic numbers

EDITs - extension to alternating sums (6) - extension to general sums with parameter $x$ (8) - extension to general sums with two parameters (11) Extended post The relation $$\sum _{k=1}^{n } \left(...
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2answers
296 views

On twisted Euler sums

An interesting investigation started here and it showed that $$ \sum_{k\geq 1}\left(\zeta(m)-H_{k}^{(m)}\right)^2 $$ has a closed form in terms of values of the Riemann $\zeta$ function for any ...
1
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2answers
78 views

For given $k,N\in\mathbb{N}$, how to compute $\sum_{i=0}^Ni^k$? [duplicate]

For given $k$ and $N$, $k,N\in\mathbb{N}$, how to compute $\sum_{i=0}^Ni^k$? We have: $\sum^N_{i=0}i=\frac{N(N+1)}2$ Also according to what I found in the Internet we have $\sum^N_{i=0}i^2=\frac{n(n+...
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3answers
260 views

Closed form for the harmonic approximation sum $\sum _{k=1}^{\infty } \left(H_k^{(2)}-\zeta (2)\right){}^2$

Question Is there a closed form of this harmonic approximation sum $$s=\sum _{k=1}^{\infty } \left(H_k^{(2)}-\zeta (2)\right){}^2\tag{1}$$ The notation is standard. Motivation This question ...
2
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0answers
68 views

Partial sum involving harmonic numbers

QUESTION: I need to know how to compute the partial sum $$\sum_{k=1}^n \frac{H_{k+1}^2-H_{k+1}^{(2)}}{k+2}$$ in terms of the generalized harmonic numbers $H_n^{(m)}$. CONTEXT: This problem arose ...
1
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2answers
66 views

Why can we maximize $\frac{k-1}{n}(\mathrm{H}_{n-1}-\mathrm{H}_{k-2})$ for $2\leq k\leq n-1$ by substituting $\mathrm{H}_l$ with $\ln(l)$?

During an excercise session in a basic course of probability it was shown that the secretary problem can be reduced to solving the following task: For a given natural $n$ optimize $2\leq k\leq n-1$ so ...
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0answers
61 views

Generating function for harmonic number times log ($H_k log(k)$)

During my studies of generating functions for expressions composed of harmonic numbers and its asyptotic approximations I found a simple sum which seems to have been neglected so far $$s171219a=\sum ...
4
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1answer
144 views

A closed form of the family of series $\sum _{k=1}^{\infty } \frac{\left(H_k\right){}^m-(\log (k)+\gamma )^m}{k}$ for $m\ge 1$

Introduction Inspired by the work of Olivier Oloa [1] and the question of Vladimir Reshetnikov in a comment I succeeded in calculating the closed form of the sum $$s_m = \sum _{k=1}^{\infty } \frac{\...
6
votes
3answers
313 views

Prove series form of fractional harmonic numbers

Let $H_\alpha$ be the $\alpha$th fractional harmonic number so that $$ H_\alpha = \int_0^1 \frac{1-x^\alpha}{1-x}\,\text dx. $$ I want to directly show $$ H_\alpha = \sum_{k=1}^\infty \frac{\alpha}{k(...
1
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1answer
38 views

How can we prove $\sum\limits_{n=1}^{k}\frac{f(m,n)}{n}\approx\frac{\varphi(m)}{m}(\ln(k\prod\limits_{p|m}p^{\frac{1}{p-1}})+\gamma)$?

First we need to create a function $$f(m,n)=\begin{cases} 1&\text{if $m,n$ - coprime}\\ 0&\text{otherwise}\\ \end{cases}$$ Then we can be sure, that for large $k$ and $|\mu(m)|=1$ $$\sum\...
1
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3answers
99 views

Sums and harmonic series

I found the solution of series on Wolfram Alpha http://www.wolframalpha.com/input/?i=sum+1%2F(2k%2B1)%2F(2k%2B2)+from+1+to+n $ \sum\limits_{k=1}^{n} \left(\frac{1}{2k+1} - \frac{1}{2k+2}\right) = \...
0
votes
3answers
89 views

Harmonic Number for $H_{n+1}$ and $H_{n+\frac{1}{2}}$

According to the definition of harmonic number $H_n = \sum\limits_{k=1}^n\frac{1}{k}$. How we can define $H_{n+1}$ and $H_{n+\frac{1}{2}}$?
3
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3answers
78 views

On a series involving harmonic numbers

Finding $$\sum^{\infty}_{n=1}\bigg[\bigg(\frac{(-1)^{n+1}}{n+1}\bigg)\bigg(\sum^{n}_{r=1}\frac{1}{r}\bigg)\bigg]$$ Try: Let $\displaystyle H_{n} =\sum^{n}_{r=1}\frac{1}{r},$ then series $\...
3
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0answers
72 views

Closed expression for almost-elliptic theta $\sum _{k=1}^{\infty } \frac{1}{2^{k^2} k}$?

Motivation: A more complicated generalization of Closed expressions for harmonic-like multiple sums are sums of a type for which the follwing is the simplest case $$s=\sum _{m=1}^{\infty } \sum _{n=...