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.

4
votes
2answers
538 views

Harmonic number divided by n [duplicate]

How do I prove that $\dfrac{H_n}{n}$ (where $H_n$ is a harmonic number) converges to $0$, as $n \to \infty$?
1
vote
3answers
278 views

Using Riemann sums to show that $\sum_{i=1}^n 1/i = \log{n} +c+O(1/n)$

I want to show that there exists a constant $c$ such that: $$ \sum_{i=1}^n 1/i = \log{n} +c+O(1/n) $$ I am thinking about Riemann sums. Any hints on that?
1
vote
0answers
635 views

Explicit formula for inverse function for n-th partial sum of harmonic series

Originally, I was trying to work out an explicit formula for a function with the property that $f(x + \frac{1}{f(x)}) = f(x) + k$ for some $k$. This is inspired by a thing I did with a metronome ...
6
votes
2answers
87 views

How to prove that for $n \in \mathbb{N}$ we have $\sum_{k=2}^n \frac{1}{k}\leq \ln(n) \leq \sum_{k=1}^{n-1} \frac{1}{k}$

Prove that for $n \in \mathbb{N}$ we have $\sum_{k=2}^n \frac{1}{k}\leq \ln(n) \leq \sum_{k=1}^{n-1} \frac{1}{k}$ by using Riemann integral?
10
votes
0answers
703 views

stirling numbers and harmonic number identities

Permit me a brief introduction before I state the question, three questions in fact. Inspired by this MSE link I computed the following harmonic sum identities: $$1/6\, \left( {H_{{n}}}^{(1)} \right) ...
1
vote
1answer
108 views

Is this formula for the harmonic numbers true?

Is this formula for the harmonic numbers true? $$H_n = \lim_{s\to 0} \, \int \frac{(s+1)^{(-n-1)}+s-1}{s} \, ds$$ Mathematica: ...
5
votes
2answers
185 views

Inequality with $\frac{1}{n}+\frac{1}{n+1}+\cdots+\frac{1}{2n}$

Inspired by this recent question, I suggest this. Let $n=2,3,4, \ldots .$ Then $$ \frac{7}{12} < \cfrac 1 {1 + \cfrac {1^2} {1 + \cfrac {2^2} {\ddots + \cfrac \vdots { 1 + \, {n^2} \,}}}} \leq \...
1
vote
1answer
282 views

Showing that $1-1/2+ \cdots +1/(2n-1)-1/2n$ is equal to $1/(n+1)+1/(n+2)+ \cdots +1/(2n)$

$1-1/2+1/3-1/4+ \cdots +1/(2n-1)-1/2n=1/(n+1)+1/(n+2)+ \cdots +1/2n$ I was asked to prove by mathematical induction the validity of the above equation. It isn't hard to prove that it holds for any ...
6
votes
8answers
262 views

Show that the inequality holds $\frac{1}{n}+\frac{1}{n+1}+…+\frac{1}{2n}\ge\frac{7}{12}$

We have to show that: $\displaystyle\frac{1}{n}+\frac{1}{n+1}+...+\frac{1}{2n}\ge\frac{7}{12}$ To be honest I don't have idea how to deal with it. I only suspect there will be need to consider two ...
1
vote
2answers
307 views

Induction inequality on sum of reciprocals

I have to prove that: $\displaystyle\frac{1}{n}+\frac{1}{n+1}+...+\frac{1}{2n}\ge\frac{1}{2}$ for natural $n$ Checking for $n=1$ we have $\displaystyle 1+\frac{1}{2}=\frac{3}{2}\ge \frac{1}{2}$ ...
14
votes
8answers
3k views

prove $\frac{1}{ n+1}+\frac{1}{ n+2}+\cdots+\frac{1}{2n}<\frac{25}{36}$ by mathematical induction

How to prove $$\frac{1}{ n+1}+\frac{1}{ n+2}+\cdots+\frac{1}{2n}<\frac{25}{36}$$ by Mathematical induction,n$\ge $1
13
votes
3answers
1k views

Harmonic Numbers series I

Can it be shown that \begin{align} \sum_{n=1}^{\infty} \binom{2n}{n} \ \frac{H_{n+1}}{n+1} \ \left(\frac{3}{16}\right)^{n} = \frac{5}{3} + \frac{8}{3} \ \ln 2 - \frac{8}{3} \ \ln 3 \end{align} where $...
1
vote
1answer
99 views

The relationship between each harmonic numbers

In Knuth's "Concrete Mathematics" in chapter about numbers below equality is given $$H_n = \ln n + \gamma + \frac{1}{2n} - \frac{1}{12n^2} + \frac{\epsilon_n}{120n^4} $$ where $0 < \epsilon_n < ...
2
votes
3answers
3k views

Convergence N'th Harmonic number minus the Natural Logarithm of N. [duplicate]

I was hoping if someone could show me the proof of exactly why this converges to the Euler–Mascheroni constant.
0
votes
2answers
124 views

Limit of a harmonic subseries minus “its” logarithm

$\displaystyle \lim_{n \to \infty} \sum_{k=1}^n \frac{1}{3k-1} - \frac{1}{3}\ln(n)$ I think that inserting the other terms and then subtracting them would not help. I need just the ideea. Thank you.
1
vote
3answers
137 views

Prove that $s_n \leq 1+\ln n$, where $s_n$ is the $n$th partial sum of the harmonic series

This is a very Interesting question, there are many ways to do it. Lets see what is the best way to do it. I have an idea which involves a definite integral, I am working on it, will post it soon.
9
votes
4answers
3k views

Showing inequality for harmonic series.

I want to show that $$\log N<\sum_{n=1}^{N}\frac{1}{n}<1+\log N.$$ But I don't know how to show this.
1
vote
2answers
111 views

Divergence of modified harmonic series

I am reading a paper which claims that the following series diverges: $\sum\limits_{n=2}^{\infty}\frac{1}{nH_{n-1}}$ where $H_{n}$ is the $n$'th harmonic number $\sum\limits_{m=1}^{n}\frac{1}{m}$. I ...
0
votes
0answers
88 views

Wolstenholme Number

Does Wolstenholme Numbers have perfect squares other than 1 and 49? The first few are 1, 5, 49, 205, 5269, 5369, 266681, 1077749 seems to be a complicated problem
2
votes
0answers
201 views

Prime Zeta Function

Does $$\sum_{p \text{ prime}} \frac{1}{p^s} \sim \log \zeta(s) \quad \text{as} \quad s \to 1^+$$ imply $$\sum_{p \leq n} \frac{1}{p} \sim \log H_n \quad \text{as} \quad n \to \infty,$$ where $H_n$ is ...
1
vote
2answers
93 views

Convergence of $\sum_{n=1}^{\infty}(\frac{H_n}{p_n}-\frac{n}{n^n})$

Does this diverge or converge ?? $$\sum_{n=1}^{\infty}(\frac{H_n}{p_n}-\frac{n}{n^n})$$ where $H_n$ is the nth harmonic number, $p_n$ is the nth prime. My impression is that it diverges, but I don't ...
6
votes
3answers
569 views

How to find the limit of a sum of reciprocals $\lim_{n\to\infty}(1 + \frac{1}{2} + \frac{1}{3} + \cdots+ \frac{1}{n})$? [duplicate]

There's a limit that I am unable to solve. I think it should be equal to $\infty$. $$\lim_{n\to\infty}\left(1 + \frac{1}{2} + \frac{1}{3} + \cdots+ \frac{1}{n}\right)$$
0
votes
1answer
120 views

A formula for $\sum^n_{i=1}(1+1/n)$?

Find a formula for $$\sum^n_{i=1}\left(1 + \dfrac{1}{n}\right)$$ Prove that it holds for all $n \geq 1$. It kind of looks like is a series but I didn't succeed in this problem. Can someone help me ...
0
votes
0answers
38 views

Series $1 , 1+\frac{1}{2}, 1+\frac{1}{2}+\frac{1}{3}, 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}$ [duplicate]

I've got this question here where I'm asked to find the limit of a sequence. But my sequence is : $$1 , 1+\frac{1}{2}, 1+\frac{1}{2}+\frac{1}{3}, 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}$$ and so on... ...
3
votes
2answers
200 views

How to calculate $\lim_{n \to \infty} \frac 1{3n} +\frac 1{3n+1}+\cdots+\frac 1{4n}$?

Could you please help me calculate this limit: $\lim_{n \to \infty} \frac 1{3n} +\frac 1{3n+1}+\cdots+\frac 1{4n}$. My best try is : $\lim_{n \to \infty} \frac 1{3n} +\frac 1{3n+1}+\cdots+\frac 1{4n}...
3
votes
2answers
123 views

Limit and sum question [closed]

Solve these limits. $$\lim_{n\to\infty}\left(\,\frac{1}{n}+\frac{1}{n+1}+\ldots+\frac{1}{2n}\,\right)=?$$ and $\lim_{n \to \infty}a_{n} = ?$ where $$a_{0}=1\,,\quad a_{n}=\frac{1}{2}\left(\,\frac{2}...
1
vote
3answers
120 views

Using $S_n = \sum_{k=1}^{n}H_k$ where $H_k$ are the harmonic numbers, show $S_n = (n+1)H_n - n$ [duplicate]

The question: Using $S_n = \sum_{k=1}^{n}H_k$ where $H_k$ are the harmonic numbers, show $S_n = (n+1)H_n - n$. So far I have $S_n = \sum_{k=1}^{n} H_k = \sum_{k=1}^{n} \sum_{j=1}^{k}\frac{1}{j} $...
4
votes
0answers
104 views

Is there an infinite hierarchy of sequences whose reciprocals diverge, starting with the natural numbers?

It is well known that the sum of the reciprocals of the function $f_0(n)=n$ (the harmonic series) diverges: $$\sum_{n=1}^\infty\,\frac{1}{n}=\infty$$ Similarly, the sum of the reciprocals of the ...
-2
votes
2answers
62 views

Show that there is a real positive solution [closed]

Show that there exists a real positive $n$ such that $(n+1) \left (1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}\cdots\dfrac{1}{n+1} \right ) \geq 5280n$
6
votes
2answers
220 views

Showing that $\sum_{i=1}^n \frac{1}{i} \geq \log{n}$

I have been trying to prove this by induction on $n\in \mathbb{N}$, but this approach seemed to get me nowhere. I have a suspicion it might be necessary to express $\log{n}$ as $\int_1^n 1/x\text{ }dx$...
1
vote
1answer
112 views

Clusters of Cars

So the original problem is: $N$ cars, each having a different speed, are going along a one-lane road, so no passing is possible. Eventually, the cars will accumulate in packets, with the "fast" ...
23
votes
4answers
758 views

How find this sum $I_n=\sum_{k=0}^{n}\frac{H_{k+1}H_{n-k+1}}{k+2}$

$$I_n=\sum_{k=0}^{n}\dfrac{H_{k+1}H_{n-k+1}}{k+2}$$ where $$H_{n}=1+\dfrac{1}{2}+\cdots+\dfrac{1}{n}$$ my try:since $$I_n=\dfrac{1+\dfrac{1}{2}+\cdots+\dfrac{1}{n+1}}{2}+\dfrac{\left(1+\dfrac{1}{2}\...
2
votes
2answers
384 views

Inequality involving sums of reciprocals and n-th root

I'm trying to prove this inequality. Let n be a positive integer. Prove that: $$\frac{1}{n}+\frac{1}{n+1}+\dots+\frac{1}{2n-1}\ge n\sqrt[n]{2}-n$$ I've tried doing it with algebraic, geometric, and ...
2
votes
1answer
440 views

Difference between two harmonic numbers when n is not infinite

I need to compute the difference between two harmonic number, in particular : $H_n - H_{n-pn}$ and i see in this answer Calculating/Estimating difference between Harmonic numbers that $H_n = ln(n) +...
0
votes
1answer
244 views

Dirichlet triangle mesh

I was reading up on the Dirichlet problem, and was truly hoping if anyone here has the time to help make me understand this a bit better. In particular, the question relates to harmonic maps. My ...
0
votes
1answer
59 views

Is it true that n is always greater than or equal to harmonic_number(n)?

My question is: Prove or disprove: For any number $n\in \mathbb{Z}$ and $n > 0$, $$n \ge H_n = \sum_{k=1}^n \frac{1}{k}.$$
1
vote
1answer
564 views

Solving an equation involving floor/ceiling as a summation bound

Is it possible to solve the following equation for $\alpha$? $$ M = \lfloor \alpha \rfloor + \alpha \sum_{k=\lfloor \alpha \rfloor +1}^N \frac{1}{k}$$ where $\alpha \geq 1$. Intuitively, $M$ is a ...
1
vote
1answer
102 views

dividing an octave to $7$ instead of $12$

Usually an octave is divided into $12$ parts based on the harmonic series(basic zeta function). how can I calculate the frequency of a note if I divide the octave into $7$ parts? $N_1=A_4(440Hz)$ ...
2
votes
1answer
98 views

Finding a sum using $\frac{sin(x)}{x}$ taylor series and in terms of its roots

I saw Euler's proof of how the sum of the squares of the harmonic numbers equals $\frac {\pi^2}{6}$ and wanted to see if it works for the $x^4$ term of the taylor expansion as well. So I found that ...
21
votes
1answer
55k views

Is there a partial sum formula for the Harmonic Series? [duplicate]

There is a partial sum formula for $$\sum_{x=1}^n x^1 = \frac{n(n+1)}{2}$$ and even one when the exponent of $x$ is $0$: $$\sum_{x=1}^n x^0 = n$$ but I cannot find one for exponent $-1$: $$\sum_{x=1}^...
4
votes
5answers
312 views

$\sum_{k=1}^nH_k = (n+1)H_n-n$. Why?

This is motivated by my answer to this question. The Wikipedia entry on harmonic numbers gives the following identity: $$ \sum_{k=1}^nH_k=(n+1)H_n-n $$ Why is this? Note that I don't just ...