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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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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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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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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 $60$ ...
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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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5answers
242 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 + \...
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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 ...
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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
109 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}{...
1
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1answer
55 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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2answers
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Asymptotic estimate of the sequence of harmonic series $\sum_{k=1}^{n} \frac{1}{k} $ [closed]

Asymptotic estimate of the sequence of partial sums, for $n\rightarrow \infty$: $$ s_n=\sum_{k=1}^{n} \frac{1}{k} $$
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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 ...
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1answer
2k views

Formula for the harmonic series $H_n = \sum_{k=1}^n 1/k$ due to Gregorio Fontana.

My question was inspired by this stackexchange question. For the last 90 minutes I have been trying to prove this formula due to Gregorio Fontana: $$H_n = \gamma + \log n + {1 \over 2n} - \sum_{k=2}^\...
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2answers
586 views

Sum of powers of Harmonic Numbers

This is a natural extension of the question Sum of Squares of Harmonic Numbers. I became interested in this question while studying the problem A closed form of $\sum_{n=1}^\infty\left[ H_n^2-\left(\...
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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 ...
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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 ...
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1answer
83 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
111 views

how to get $\sum_{k=1}^\infty \arctan\biggr(\frac{10k}{(3k^2+2)(9k^2-1)}\biggr)=\log3-\frac{\pi}{4}$

problem in the above asked equation of S.Ramanujan ! Hello everyone,this is a result of an entry described by ramanujan,i first request you to see the photo i have attached. MY ATTEMPTION From LHS ...
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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 ...
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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}+...
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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 ...
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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 ...
27
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2answers
844 views

How find this series $\sum_{n=1}^{\infty}\frac{1}{n^2H_{n}}$?

Question: This follow series have simple closed form? $$\sum_{n=1}^{\infty}\dfrac{1}{n^2H_{n}}$$ where $$H_{n}=1+\dfrac{1}{2}+\dfrac{1}{3}+\cdots+\dfrac{1}{n}$$ I suddenly thought of this ...
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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 ...
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1answer
100 views

Exact formula or approximation for this sum (general harmonic series $H_{n,3}$)

I encountered the following problem in my studies. I want to calculate the requirement to the parameter $a$ for a local minimum in the function: $F(N;a) = -a*(N-1) + \sum_i^{N-1}\sum_j^i \frac{1}{j^3}...
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1answer
383 views

A Summability methods which sum the harmonic series

Studying summability theory I've come across many summation methods however by now I know only two not very interesting method which re-sums the harmonic series $\sum_{n=0}^\infty \frac 1{n+1}$ : the ...
4
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1answer
99 views

Reciprocity of different prime numbers can approximate $1$?

I want to see if there exist $p_1<p_2<p_3<\cdots<p_{1000}$ different prime numbers such that $|1/p_1+\cdots+1/p_{1000}-1|\le ({1\over p_{1000}})^2.$ a) what is my point with this? Nothing....
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0answers
38 views

A simpler definition of the Harmonic number asymptotic expansion

I know that $$H_n = \ln n + \gamma + \frac{1}{2n} - \sum_{k=1}^\infty \frac{B_{2k}}{2k}\left(\frac{1}{n^2}\right)^k$$ I am thinking this could be simplified to $$H_n = \sum_{k=0}^\infty a_k\left(\...
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2answers
74 views

Lerch transcendent and harmonic number simplification

I am trying to find a simplification of the following, preferably to a hypergeometric function. I have the result in Mathematica notation: ...
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0answers
46 views

Infinite/Recursive Cesàro Summation of $\zeta(1)$

Is anything known about this kind of `infinite' Cesàro summation (or any related types of summation)? If we have a function we wish to sum $f(n)$, but $$ S^0[f] = \sum_{n=1}^\infty f(n) $$ diverges, ...
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2answers
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 ...
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 ...
3
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1answer
56 views

$\lim_{N\rightarrow\infty}\left< \psi{\left(N_k+1\right)} - \psi{\left(N_j+1\right)} \right >$

Question What is \begin{align} \lim_{N\rightarrow\infty}\left< \psi{\left(N_k+1\right)} - \psi{\left(N_j+1\right)} \right >? \end{align} Here, $N_k$ and $N_j$ are multinomially distributed ...
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} + \...
3
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4answers
296 views

Test for the convergence of the sequence $S_n =\frac1n \left(1 + \frac{1}{2} + \frac{1}{3} + \cdots+ \frac{1}{n}\right)$

$$S_n =\frac1n \left(1 + \frac{1}{2} + \frac{1}{3} + \cdots+ \frac{1}{n}\right)$$ Show the convergence of $S_n$ (the method of difference more preferably) I just began treating sequences in school, ...
1
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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 \...
2
votes
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 ...
4
votes
2answers
205 views

Generating function of square harmonic numbers

During my study of generating functions, I was able to calculate the generating function of the sequence of harmonic numbers $H_n$: $$\sum_{n=1}^\infty H_nx^n=\frac{\ln(1-x)}{x-1}$$ However, I also ...
4
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0answers
2k views

Problem with the expectation of a maximum of independent gamma distributed random variables

Having a problem with the expectation of the maximum among $n$ independent random variables $ X_1, X_2 \dots X_n$ all ~ the same class of distributions but not necessarily the same mean and other ...
1
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2answers
112 views

$\sum_{n = 1}^\infty (-1)^n \frac{H_n}{n^s}$ in terms of $\sum_{n = 1}^\infty \frac{H_n}{n^s}$

I have been looking for an equation that relates the above sums to no avail. Perhaps, I am missing some important Harmonic identities. In the sums, $H_n$ represents the $n^{th}$ harmonic number. ...
2
votes
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 ...
2
votes
2answers
130 views

Solving limit $\lim_{n\rightarrow \infty} (\frac{1}{n+1}+ … +\frac{1}{2n})$ [duplicate]

Could anyone help me solving the following limit? $\lim_{n\rightarrow \infty} (\frac{1}{n+1}+ ... +\frac{1}{2n})$ I think it should just be 0 since we can distribute the limite inside the sum, but ...
-1
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7answers
142 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 ...