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.

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
votes
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 ...
6
votes
8answers
477 views

How to show that $\lim \frac{1}{n} \sum_{i=1}^n \frac{1}{i}=0 $? [duplicate]

Show that $$\lim \frac{1}{n} \sum_{i=1}^n \frac{1}{i} =0 $$ I've proved that this sequence converges (it is bounded and decreasing). NOW, I need to find a sequence that is bigger than this one and ...
6
votes
4answers
173 views

Prove that $1<\frac{1}{n+1}+\frac{1}{n+2}+…+\frac{1}{3n+1}$

Prove that $1<\dfrac{1}{n+1}+\dfrac{1}{n+2}+...+\dfrac{1}{3n+1}$. By using the Mathematical induction. Suppose the statement holds for $n=k$. Then for $n=k+1$. We have $\dfrac{1}{k+2}+\dfrac{1}{k+...
1
vote
3answers
150 views

Mathematical induction for inequalities: $\frac1{n+1} + \frac1{n+2} + \cdots +\frac1{3n+1} > 1$

Prove by induction: $$\frac1{n+1} + \frac1{n+2} + \cdots +\frac1{3n+1} > 1$$ adding $1/(3m+4)$ as the next $m+1$ value proves pretty fruitless. Can I make some simplifications in the inequality ...
1
vote
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 $$\sum_{k=1}^{n}\frac1{k^2}$$
1
vote
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 ...
0
votes
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 $$\left(\frac{1}{y} + \frac{1}{z}\right)\left(\frac{1}{z} + \frac{1}{x}\right)\left(\frac{1}{x} + \frac{1}{y}\right) \geq \frac{8}{xyz}$$ ...
0
votes
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&...
-1
votes
2answers
339 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 $...
0
votes
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
votes
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 ...
9
votes
4answers
610 views

Proving Binomial Identity without calculus

How to establish the following identities without the help of calculus: For positive integer $n, $ $$\sum_{1\le r\le n}\frac{(-1)^{r-1}\binom nr}r=\sum_{1\le r\le n}\frac1r $$ and $$\sum_{0\le r\le ...
2
votes
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{...
9
votes
2answers
345 views

Is there a closed form for the alternating series of inverse harmonic numbers?

Let $H_n=\sum _{k=1}^n \frac{1}{k}$ be the n-th harmonic number. Since $H_{k+1}>H_k$ for $k=1, 2, 3, ...$ the sequence $\frac{1}{H_k}$ is monotonic decreasing as $n \to \infty$, and the Leibniz ...
43
votes
5answers
8k views

Do harmonic numbers have a “closed-form” expression?

One of the joys of high-school mathematics is summing a complicated series to get a “closed-form” expression. And of course many of us have tried summing the harmonic series $H_n =\sum \limits_{k \leq ...
2
votes
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/ ...
11
votes
5answers
10k views

Proving $\frac{1}{n+1} + \frac{1}{n+2}+\cdots+\frac{1}{2n} > \frac{13}{24}$ for $n>1,n\in\Bbb N$ by Induction

Proving $\frac{1}{n+1} + \frac{1}{n+2}+\cdots+\frac{1}{2n} > \frac{13}{24}$ for $n>1,n\in\Bbb N$ To solve it I used induction but it is leading me nowhere my attempt was as follows: Lets ...
11
votes
3answers
413 views

Is the following Harmonic Number Identity true?

Is the following identity true? $$ \sum_{n=1}^\infty \frac{H_nx^n}{n^3} = \frac12\zeta(3)\ln x-\frac18\ln^2x\ln^2(1-x)+\frac12\ln x\left[\sum_{n=1}^\infty\frac{H_{n} x^{n}}{n^2}-\operatorname{Li}_3(...
22
votes
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: ...
1
vote
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)...
8
votes
6answers
282 views

computing the limit of $\frac{1}{n} \sum_{k=1}^{n}{\frac{1}{k}}$ [duplicate]

Prove that the following limit is 0 $\lim\limits_{n\to\infty} \frac{1}{n} \sum_{k=1}^{n}{\frac{1}{k}}$ Please I don't know how to do it :S. The harmonic series diverges thus it requires some special ...
3
votes
4answers
676 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....
3
votes
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 ...
1
vote
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}^\...
0
votes
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
votes
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 $\...
2
votes
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}^...
0
votes
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 ...
2
votes
0answers
65 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 $\zeta(s)$...
1
vote
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$...
1
vote
1answer
148 views

Double series of Harmonic Numbers

In a solution presented here a series involving the product of Harmonic numbers is involved. The intent of the problem is to determine a form of the series \begin{align} \sum_{n=1}^{\infty} \frac{H_{n}...
2
votes
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 $\ln k$, ...
-4
votes
2answers
66 views

Find the value of: $\lim_{n \to \infty}\left(\frac{1}{n+1}+\cdots+\frac{1}{2n}\right)$ [duplicate]

Calculate $$\lim_{n \to \infty}\left(\dfrac{1}{n+1}+\cdots+\dfrac{1}{2n}\right).$$ I tried relating it to a Riemann sum, but I couldn't see how to do so.
1
vote
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 ...
1
vote
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 ...
1
vote
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 ...
2
votes
1answer
113 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}{...
3
votes
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=...
2
votes
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 ...
9
votes
3answers
244 views

Existence of a sequence $\{\epsilon_n\}_{n\ge 1}$ such that $\sum\limits_{n=1}^{\infty}\frac{1}{n^{\varepsilon_n}} $ converges

I am trying to find a sequence $\{\varepsilon_n\}_{n\ge 1}$ such that $$\lim_{n\to \infty}\varepsilon_n=1~~~~~~~~~~~~~\text{and}~~~~~~~~~\sum_{n=1}^{\infty}\frac{1}{n^{\varepsilon_n}} <\infty.$$ ...
1
vote
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 ...
1
vote
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(...
-1
votes
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^...
13
votes
2answers
297 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
vote
3answers
135 views

Finding the general/closed form of $\sum_{k=1}^n k^a$ [duplicate]

I recently noticed the following: $$\sum_{i=1}^ni=\frac{n(n+1)}{2}$$ $$\sum_{i=1}^{n}i^2 = \dfrac{n(n+1)(2n+1)}{6} = \Bigg(\sum_{i=1}^ni\Bigg) \cdot \frac{2n+1}{3}$$ $$\sum\limits_{i = 1}^n i^3 = \...
1
vote
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+...
1
vote
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 ...
2
votes
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
vote
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 ...