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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.

14
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1answer
189 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
65 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
35 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
57 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
votes
3answers
146 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: $$1+\frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n}$$ To my surprise ...
2
votes
3answers
86 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
4 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
85 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
22 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
47 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
39 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
40 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 ...
2
votes
3answers
124 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
25 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
77 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
70 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}{...
1
vote
1answer
43 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 ...
2
votes
0answers
51 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}...
0
votes
0answers
37 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}...
1
vote
0answers
54 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
votes
5answers
211 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 + \...
2
votes
0answers
70 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
vote
1answer
66 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 ...
0
votes
1answer
83 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
votes
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
votes
2answers
121 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
votes
0answers
28 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
votes
1answer
50 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 ...
0
votes
1answer
49 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}...
4
votes
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....
0
votes
0answers
36 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(\...
1
vote
2answers
54 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: ...
0
votes
0answers
37 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, ...
2
votes
1answer
95 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 ...
1
vote
2answers
147 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
votes
2answers
74 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
votes
0answers
80 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 ...
0
votes
0answers
26 views

Does the harmonic sum “lands” on whole number? [duplicate]

Since the harmonic sum diverges, it clearly passes through infinitely many whole numbers, but i was wandering if it "lands" on one: is there a number $n>1$, so that: $\sum_{k=1}^n \frac{1}{k}$ is a ...
2
votes
4answers
100 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
votes
0answers
53 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 ...
-1
votes
7answers
137 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 ...
3
votes
3answers
116 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 \...
1
vote
2answers
65 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 $$\sum_{k=1}^{n}\frac1{k^2}$$
1
vote
2answers
51 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 ...
0
votes
2answers
70 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 $$\left(\frac{1}{y} + \frac{1}{z}\right)\left(\frac{1}{z} + \frac{1}{x}\right)\left(\frac{1}{x} + \frac{1}{y}\right) \geq \frac{8}{xyz}$$ ...
0
votes
1answer
52 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&...
0
votes
2answers
133 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
votes
2answers
99 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 ...
2
votes
2answers
114 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{...